Part of me thinks it would be helpful for him to rent someplace ... but he keeps telling me that's like giving up on the hope we can get past this
So, turning this on its head, it seems as though he feels that if he does move to the new house with you, then that is giving him the signal that you will get past this - and you are not ready to give him this signal.
Ask him if the shoe was on the other foot, could he ever forgive such a betrayal?
I wouldn't bother asking him this - he'll just say yes (even if it isn't true).
I know he has done a lot of removal/packing work - but he would have been doing that anyway. He has been reluctant and/or slow to do the things you have asked him to do to show that he is serious about making amends - coming off his game, deleting people etc - he had to be given time to do that (and I suspect his parents put their oar in there too).
I can tell though from the tone of your posts that you are very very weary, so if it works to have him there for a pre-arranged limited amount of time, then I am sure nobody would blame you - even if then the limited time was perhaps extended. I think it will be hard to then say, after 3 months or so that no, it isn't working, you can't get over it and he needs to leave - but this possibility will have to be made clear to him and he must expect it.
You ask how people knew what to do in these circumstances. Well, like you, they don't always. My ex had an affair when we were both coming up to 40. He wanted to leave with the OW and move abroad with her and leave me with our children who were then 10 and 8 - but at the last moment she changed her mind, dumped him and went back to her ex. My ex was devastated at losing her and having ugly old me instead (she was 10 years younger).
I wanted my him to leave - but like yours, he didn't want to. He didn't have a job, he was doing final exams blah de blah. He refused to live with or even tell his parents. He got angry with me when he kept asking me what I wanted and I kept saying that I wanted space, for him to leave. In the end, in temper, I said "I keep telling you what I want and you keep telling me I can't have it - so why don't you just tell me what you're going to do?". He then asked for another chance and I was weary and sad so I said yes.
Eight years later HE decided it wasn't what he wanted and buggered off anyway - at a much less favourable time for me. I kept his dirty secret for 8 years - and what good did it do me? I have never had another partner - and not only that, neither of our children (who are now in their 30s) has a LT partner. I am sure that somehow, even though we didn't intend it, our relationship damaged them in this way. I now consider those eight years to have been wasted from my point of view.
You aren't in quite so bad a position as I was - family and friends know what he's done, so they don't expect you to be playing happy families. You sound as though you have a reasonable income and you have supportive family (well I had that too) - but you have to make it clear to him that you are not in a position to decide for certain now whether you want him back.
I am really sorry this is so long - hope it is of some use to you.