I would definitely refuse to come over unless he's leashed, I'm shocked they aren't already doing this. Is this just with you or do they leave him loose when anyone comes over? I know with my dog entrances are by far the most dangerous (or would be if he were not leashed). For him that is the only time he would bite without the person trying to interact with him. He is just soooooo over the top aroused that he isn't thinking at all. Even when I have people over he knows and LIKES I leash him when they walk through the door unless it is someone who comes over all the time and he can recognize by sight. With less frequent visitors he often doesn't recognize them until he sniffs, so he may behave aggressively for a few seconds until he realizes who it is. Especially if they knock, he believes knocking is something strangers, not friends, do so a knock puts him straight into aggressive mode and a bite could happen before he knows who he's sinking his teeth into. Your mom must understand that the reason for the dog's behavior doesn't matter, you feel unsafe (because you are) and that should be enough reason to put a leash on the dog.
You are right that a muzzle does not fix the problem, it's a tool that can be used in addition to a leash to make things safer just in case. For me a muzzle would be used only if the dog has shown signs of warming up to someone and your mom wants to let him sniff the person. A muzzle would make it possible for the dog to truly get comfortable with the person on a "normal dog" level (actual interaction). Without a muzzle I wouldn't let the dog within a few feet of anyone, especially since he sounds very easy to set off.
As for actual training she should do, it's really hard to say without seeing the dog. Different things work for different dogs. I can only tell you what I do with my own dog, who sounds very similar to Benny, albeit with better bite inhibition. Does his reactivity decrease at all during your visit (does he start super bad, reacting when you do barely anything and then get less easily upset throughout the visit?).
So with my dog Tucker He is leashed when someone comes over and I usually take him into a room where he cannot watch the people enter. he will still react because he can hear them, but it's not as bad as when he can see them too. It's even better if we can open the door before anyone knocks, knocking makes him go nuts. If I had enough prep time I will have him on a harness, but if it was unexpected a collar works fine too. I pack my pockets with treats, lots of treats. So he barks his head off and we stand in another room and I wait until i get some sign that he can think. I usually try to get his attention with his name or a noise. If he looks at me he gets a treat. Sometimes he won't give me attention so I just start by dropping some treats on the floor. If he can break his attention on the visitors to eat them I can usually then get his attention with my voice and reward him for looking at me. You cannot really reward aggression with treats or a soothing voice. Remember it is an emotional response, the dog will stop the behavior when the emotion changes. So the best thing you can do is try and change the emotion from terrified or angry to happy or calm. Treats and focus help do that.
So once I get his attention I will walk us to a point where he can see the people. As soon as he sees them I get his attention again and give a treat. I will usually retreat a few steps back to where he cannot see them anymore and give another treat, ask for a sit or down, give another treat, then move forward to where he can see them again. We rinse and repeat until I think he can actually stand and watch them for a while. When this is all depends on the dog and the signs he's giving. For Tucker I can tell he's still ready to react if he is straining hard against the leash to the point where his nails dig into the floor when I pull him back. He will also usually make a very gasping sound, like he's struggling for air (even when on a harness), because he is in such a high level of panic.
When these signs go away we can usually stand and watch the people from a distance. I will let him look for a second and then get his attention for a treat. I will sometimes ask for a sit or down for a treat just to break his focus from the visitors for a little longer. I will still retreat to another room if I think he's getting more stressed again, or simply if we've been there a while and I want to give him a break. When he's doing well we will move into the room with the visitors and basically do the same things. Let him briefly watch, then ask for attention or commands. No punishment or raised voices, remember we want to get him calm and happy. The attention getting is just his name or a clicking noise with my mouth(though an actual command taught separately, like "watch me", could be used as well). Any sign that he wants to react and I'll retreat from the room with him, often tossing treats on the floor along the way to keep him from turning back to bark.
When he is remaining calm in the room with them I can usually sit down and lay off the treats a bit. We are out of the reaction zone, he will not be set off as easily anymore. I will now give treats occasionally for sitting or laying nicely. If he is staring at the person I will get his attention and give a treat. If he is air sniffing I will usually let him continue for several seconds before getting his attention and giving a treat. Sniffing is good. What I also sometimes do at this point is bring out a stuffed kong (with his dinner or simply yummy stuff in it) and let him have it. At this point he is calm enough to focus on the toy/food and it lets me have a chance to relax.
Eventually he just relaxes and can be normal and non-reactive, but still on leash. If the person does something "new" like coughing, laughing, standing up, etc. I will give him a treat (or several if it made him pretty uncomfortable). I will also reward him if the person looks at him or talks to him. I don't let them do that at first (I tell them not to when they come in and remind them if they slip up), but once he's calm I allow a tiny bit if he appears calm about it, I tell them to stop if he seems upset. I never allow them to try and touch him unless they've been here for several hours and he seems extremely comfortable with them/seeking their attention. I would not allow that at all if he had bad bite inhibition, and generally it doesn't happen on the first visit anyway. Many people are not allowed to touch him until they walk through the door and he seems thrilled that their here. What I do try and let him do to all visitors, once he is comfortable, is get an up close sniff. I will ask the person if it's okay and will tell them to just stay still and ignore him. Then I let him sniff their legs to his heart's content, then call him away for a treat. This relaxes him IMMENSELY. If I don't allow him an up close sniff he obsesses over the person for a lot longer. But again, this is not going to be safe for all dogs, especially without a muzzle.
So that's what we do in the house, and it works fairly well. It actually works best when we have multiple visitors and all but one or two are people he likes. So like he's now comfortable with all grandparents, and several aunts/uncles. If we have a groups over with nana, papa, my aunt Terry, my brother's girlfriend (all of whom are his buddies), and then two strangers, he is going to do MUCH better than if it were just the two strangers. He just sort of groups the strangers in with the visitors he likes and is less edgy. But for some dogs the large group would make them MORE upset, because groups are simply upsetting to them. This method has allowed him to become comfortable with all regular visitors we have. It still has to happen with every new person though. I find the more new people we have over in a short time frame (say we have three sets of new people over in the course of two weeks) the quicker each intro goes. If, on the other hand, we go a couple of months between visitors he's equally bad each time.
As you can see it takes a lot of time and effort every time there is a visitor. The way things work in my house I am not the one catering the the guests, so I can devote myself to the dog. If I was living by myself or was having someone over for ME it would be pretty rough. I'd either have to have patient guests or Tucker would just have to go stay in a room by himself for the duration.
Outside I basically do similar things. Rewarding him for attention, asking for commands and rewarding. But I have never gotten him to warm up to someone outside of the house, I just don't hang out for hours with people outside of the house and that's how long it would take him to warm up. I also always reward when we go by someone outside, even if we're not interacting with them. I want him to see approaching people as good things. Every time we pass someone on a walk Tucker immediately gets into heel position and stares up into my face because he knows he'll get a treat. It's harder outside though because if he does react while I'm talking to someone it'd be pretty rude to just walk away from them. So I still have some issues with that. I just have to try and shove treats in his face and hope he'll take them and calm down. My best bet is always to keep the rate of reinforcement high from the beginning so he doesn't react in the first place. If I reinforce a LOT at first he can usually calm down so I can get him beyond the risk of reacting and give fewer treats.
It's lots of work though, and whether or not your mom wants to do all that is up to her. What she NEEDS to do is manage him by putting him away or putting him on leash.