My ex wants to come into the family home as he pleases

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I split with my cheating ex, after a 25 year relationship, 7 months ago. I have 2 young adults living at home and 1 at uni.
Since the day we split up, we have still been a strong family unit and I have allowed him to come and go as he pleases. I know mad isn’t it, but it worked for all of us, however he’s now seeing someone else.
This hurts but I realise we’ve all got to get on. He’s being very disrespectful (texting her in front of us and saying he’s busy seeing her etc).
I’m now trying to put my foot down and telling him he can come and see the children and dog, only when I’m not here and to ring first. He says this doesn’t work for him (I’m cutting the story short) and if I won’t allow him in then we’ll have to sell the house as he wants to be able to buy a family home and not rent his poxy flat anymore.
He makes me miserable when I see him and this isn’t a good feeling in your own home.
I know I’ve played it all wrong from the beginning but didn’t think it would come to this.
Should I sell up and try to support myself without him, which will be hard, or bite the bullet and allow him in?

Once again, thanks for reply’s. I suppose it’s confirmed what I already thought and that is to sell up and move on with my life without him in it.
Just to be clear, he doesn’t want to sell as he wants me and the children to stay in the family home but as I’ve changed the goal posts and they don’t suit him, then he sees this as his only option as this will get him a family home out of his equity.
I did like him coming round as we get (got?) on well and as I said we’ve been a strong/close family, but things seem to be moving on with his new gf therefore it all seems very inappropriate that he still counts this as his home when he’s renting a perfectly decent flat.
Sometimes i dread coming round the corner incase his cars there. I can’t live like that anymore.

Yeah I agree with you OP, the best thing is to sell up and move on.

You are both single now and as such he's able to date whoever and whenever he wants, as are you.

He can see the kids at his own place, or out. They are old enough to meet up with him, it's not like he's coming round to do bath and bedtime.

I stayed in family home with DC and 4 years on I still get digs about being given a house. I paid him savings and took on the mortgage, but it's his excuse to criticise me. Moving is probably more straightforward in the long run.

Sell up
Put the kids on council waiting list as they will be home less.

At the beginning when you were transitioning out of your relationship, it probably was better for you all if he came round in an ad hoc way.
Things are different now. Sit him down, have a conversation. Do you just enter his flat? I suspect not.
Having notice, and arranging times in advance is reasonable. If he doesn't agree, then sell...

Sell, or at least try to buy him out. And do it soon, the longer he is with the girlfriend the nastier he will become.

yes you will just need to sell unfortunately. now the kids are adults it's not appropriate to maintain a joint home any more. you need a clean break.

as you never married your rights are limited, however if he is a reasonable person do a little totting up of things like the impact on your earning potential and pension of any uneven split in family responsibilities you have had over the years. unless you have been scrupulously fair throughout, it is likely that he is much better off than you in these things as a direct consequence of decisions you made together about child rearing. in a fair split you therefore should maybe get more than 50% of the house - but he would have to be really fair-minded and you would need his agreement as without marriage you have no legal recourse.