Maybe she wouldn't get a fair split of equity if he's not working but really there's nothing she can do about that if she wants to detach herself from him emotionally, which means not supporting him emotionally in any way.
A few posters have suggested he'll get spousal maintenance if he's not working, like it'll be a done deal, a clear cut case, when in reality it's nothing like that and he's highly unlikely to get it. My post that you originally quoted was in response to one of those posters (I don't know if it was you or not I was originally responding to).
I personally believe she's better off getting the house sold (or him out of it, if she can buy him out) because getting half the house and half the savings is way better than getting none of it because he's spent all the savings cocklodging with her, not looking for a job and refuses to leave the house/tries to get it all assigned to "poor, hard done by, unemployed, him" in the divorce.
I'd rather be arguing over who has what money from pensions and sale of house, whilst I lived in either the family home or a rental wherever with my child and without him, than risk a situation of having to live with him for X amount of years whilst he does everything possible to stall the sale of the house as punishment to OP for not rolling over and accepting being cheated on and him cocklodging.
Of course OP needs legal advice before she does anything concrete, but with 50/50 assets as a starting point for negotiation, I'd be talking half the savings today, ensuring he can't run up debts on any joint accounts as a matter of urgency. Also hunting for financial information, as she is doing and I'd be getting the house valued as a minimum, followed by talking to the mortgage provider about her ability (or otherwise) to buy him out. Just so she's got some idea where she stands. She's already looking at increasing her income so that's good for her future survival.
I'd be working from the proviso that unless a solicitor advised otherwise I'd be out of a relationship with him ASAP. I certainly wouldn't be planning on taking part in the living hell of having him hanging around for 6 months while he prevaricates over obtaining any job that's available because he's got an ego the size of the UK and believes all those jobs are beneath him, refusing to acknowledge he's got almost no chance of being employed at his previous level/career/industry.
As for his role as a father, I'd consider my child didn't need to be anywhere near such a toxic, fucked up, selfish individual and I'd sincerely hope he fucked off and never saw the child again. I'd happily accept no child maintenance for that, if not claiming it meant he fucked off quietly, instead of him arguing for 50/50 contact just so he didn't have to pay any child maintenance anyway. Co-parenting with toxic people sucks and doesn't benefit the child IMO. One stable home that the DC doesn't have to keep leaving each week and one sane reasonable parent is better than 50/50, or even EOW, with a toxic selfish parent and all the house-hopping it requires. I think often both parents like 50/50 for themselves so they can have child free time and not pay child maintenance.
IME vast majority of DC end up LC/NC with an absent parent by the time they reach their teens and have thoroughly had enough of house-hopping and being away from their friends and possessions EOW. If they continue to visit IME it's because both parents expect it and the fade of contact comes at 16/18 when child maintenance stops. Rarely does the contact on any meaningful level, or the house-hopping, continue. Once there's no more child maintenance to pay as punishment for not having the child to stay, the man and his new partner (who's usually been reluctantly putting up with it and counting down the days until it ends) loses interest in having the child to stay.