I think it depends how you're renting. If you're in social housing, as you are and like my parents were, then stick with it! As you said, you have to have extreme problems or be a very bad tenant before you can be evicted.
But a friend of mine is in privately rented property - they can't get anything on social housing round here although they are on a waiting list (that is ever growing). Both working, a couple of kids. But they are constantly worried when it comes to the annual renewal of the lease a) because the price could increase and b) because the landlord could decide to sell. Their landlord is now retired - only just so relatively young in the scheme of things. But now their concern is the future. If he dies the house goes to relatives who may decide to sell. Then they'll have to start house hunting. They get repairs done but that's at the behest of the landlord - if the bathroom taps start to leak he may not see a priority over, say, the boiler breaking down so it could take time to get things done.
We own our house so we're not in that situation. But we still had a mortgage to pay and, as you say, we had to do our own repairs. DH was made redundant about four years ago and the mortgage was the priority. It ate up all our savings and the credit cards were maxed out before he got a job so we're now paying back that debt. Mortgage is now finished thankfully so we've got enough to cover the debt and we're trying to get it down fast so we can build a small savings buffer again.
If it happened again? The house would be sold, we can downsize. But we'd probably have to cut the price to get a quick sale if we had mammoth debts I would think.
The good side of private renting and owning is that you can be flexible about where you live to a point. Obviously you have to sell in order to buy if you move to another part of the country - private renting is relatively easier then as you just have to give notice if you're on a rolling tenancy.
I know when my parents wanted to move us because of work in the 1980s they had to find someone who wanted to swap their council house in that area. I think they also went on a council waiting list but that was almost a waste of time.
I assume it's the same now. And that is a headache as you've no idea if anything is available in the area you need to move to - so it's not so flexible.
I assume you can do something via the social housing people in that situation but I imagine it's not as easy as saying 'my job's relocating to Scotland so we need to move from Wales' and just pack up and go. Any long term 'house for life' benefits would change too as it'll be a different housing provider.
My mum moved from her council house to a council flat to be nearer us when dad died and her tenancy changed to a five year renewable one from a 'home for life' one previously. So much more precarious.
So swings and roundabouts. Most people, early on in their mortgage, are pumping money into paying it off in answer to your question. It's only the latter years when the mortgage begins to look smaller compared to money coming in that that changes. And, of course, when you retire, whilst you have a house with no rent/mortgage, you're also less likely to be able to get extra cash if you need it quickly (I can't imagine many of our 80 year old neighbours having huge pensions or be able to get a bit of work if they needed to repair the roof!) But you do have an asset to sell and move into something smaller/somewhere cheaper if you're lucky.
If your DH is desperate to own, why not look at 'right to buy'? I'm not sure if it still exists but if it does it's certainly cheaper than buying on the open market.
My aunt was a council tenant for 10 years or so (I think) and got 30 ish % off as a discount. I think she got it for somewhere around £200K instead of nearly £300K - so a big saving. I presume, though, that you having been in arrears would count against you in that (but also in any mortgage to be honest - they do check).