OK. First, SAHP are no more lazy than a working parent. Lazy people are lazy people. I have friends who take (even in these times) their kids out every single day, rain or shine and friends who will happily stick the kids in front of the TV (and not just so they can use the loo, cook dinner or have a cuppa in the teeniest amount of peace). Seeing them before kids came along and most of the time you'd see a similar theme - always out and about...busy, busy, busy or happy to sit down, feet up whenever possible. Nothing wrong with either as long as the kids are OK and the home is happy.
Second, a marital/partnership home is a joint one. End of story. If my DH dies, it comes to me. If the home was owned by him alone, his family could ask me to leave if the home is left to them. If we divorce, it's half mine. I put money into it, I kept the home nice with meals on the table when I was a SAHP full time. I've contributed to making a happy family home and half of it is mine. That's it. No argument.
Third, whilst the house work is easier in some ways for a SAHP, it's not all their responsibility. DH used to take over DS when he got home from work - I have given my DS a bath about a dozen times in his life when he was a toddler. DH did that and enjoyed it - it was 'their' time, a chance to take the mickey of mum, have a chat, have a giggle, play splashing games. Learning to enjoy each others company. DH does the evening washing up and always has apart from when he worked away from home pre-Covid. He does the housework as and when needed - he'll mop the floors whilst I hoover the living room for example. It frees us both up quicker to enjoy a day out or a film night.
As far as jobs are concerned, take a look at the local schools. There are sometimes, if you're lucky, dinner ladies type jobs. Ideal to get your foot in the door. If you're known to the school they'll let you know if something else comes up. My sister started as a dinner lady then was offered a job in the finance department in her local school. Most of the holidays off bar a few days. They're like gold dust but it's worth asking and finding out where they advertise.
And, if you have time, think about volunteering. It doesn't get you money but it shows willingness and you never know where it leads. I did that when my mum was diagnosed with Alzheimer's. With 20 plus appointments to cover in a year I had no hope of getting a paying job but it helped to have that on my CV when my current (school based) job came up.
You really need to sort out the issues around the house and the amount you get. I got an allowance when DS was small and I was SAHP - but it was for me alone. DH paid the bills, paid the credit card for anything DS needed and often said 'tell me how much the coat was and I'll transfer the cash - that's too high a price out of your cash'. I think that's normal.