It's not like they had to live together full time as you said, but they could have also moved in with the grandparents with the kids for a while. So yeah they wouldn't be unsupervised, but the amnesia man could start to bond with his kids and start to feel like a part of the family. And I get that some families don't involve kids, but in this case it does and they needed to account for that. The poor kids also haven't heard from or seen their two parents, it's already hurtful for them as it is.

When I was looking at info for the latest chapters, everyone was talking only about the red head and totally ignored the side plot with the alpha guy. Like sure, he's a dick, but he's also given little patience. The man has amnesia, obviously he can't remember being all lovey-dovey with his partner. The red head isn't honest about what he feels either, the same way the alpha guy isn't honest about what bothers him. The alpha guy feels like people don't actually care about him, they just care about who he should be. Meanwhile the red head feels like his partner doesn't want him the same way anymore because he can't accept his current life. But like, he had a chance to be honest and express his feelings when the alpha guy asked why did they marry each other? Instead he says, I like your face and the sex. Yeah sure, that will definitely help the amnesia man who doesn't remember why you're together. And to have the audacity to get mad that the amnesia man doesn't remember why they got married????????? Like yeah, expect the honest truth from a man who can't remember the answer. If the goal was to just wait until the alpha guy's memories returned, then sure don't tell him anything but at least understand that he won't understand things because he doesn't remember. It would have been better to start with open feelings, leave the tsundere shit for later. And the photo, it seems like the alpha turns it down because of fear - because everyone is waiting for a better version of himself to come back. And he's had his chances to also ask questions about his past, but he's too scared. And he should've asked because he's only ended up alone because he can't understand why his actions may be misconstrued - maybe it's the fact that you haven't actually told your partner that you feel like he doesn't actually love you, but loves another version of you???? Maybe stop and talk instead of just have sex, because neither of you want to express yourselves and it would be healthier to put off the sex until recovery is done. I get the fact that he's insecure, but he should ask more questions about his past instead of waiting until his memories come back. But no, just waiting around will definitely help, right? He hasn't even mentioned how he feels sick, like what do you expect?
And why wouldn't they keep their kids around to help with the recovery???? Like I get not bringing them to the hospital, but wouldn't it make sense to have them around to help with the recovery. The family dynamic is important and the alpha guy should be a part of it to help stimulate his mind. The reason they don't feel like a family is because they aren't living as one family, they're just walking on egg shells around each other. But if the kids were around, maybe it would help the two focus on a task together on neutral grounds. But no, they push the kids out and now they have this situation.
Like they both suck in their own ways, but clearly some ppl like the red head more and give him more grace. Personally they both suck this arc.