I'm very unsure about areas being ring-fenced for locals. That would be fab if you are lucky enough to have been born in a desirable area and can stay there in social housing, preferential rents, or even sales that are blocked to 'incomers', but it's hard lines for those born in Grotsville who are unable to access any of those things in areas other than those surrounding their place of birth. People from disadvantaged areas have enough to contend with without creating systems that keep nicer areas out of reach in favour of locals.
The second home issue is a tricky one, I think. I don't think that anyone needs (or should have) a second home as long as some people don't have a first one, and I am not keen on a system that allows the 'haves' to make a living by renting to the 'have nots'. I think that the tax system should actively discourage people from having second homes, whether as holiday homes, Air B&Bs or rentals.
On the other hand, there are a lot of couples who get together with a home each, eg after divorce or bereavement, and it's difficult to know how to deal with that fairly. Whose home should be classified as the second one, and at what stage in the relationship?
I believe that all adults should have responsibility for paying tax and should be eligible for benefits based on their own contributions alone, so it wouldn't sit well with me to have a system that treats people as a couple when it comes to home ownership.
This is why I'd make a hopeless politician?. I can talk myself in and out of a school of thought so easily.
OP, my son is in a similar situation. He is hoping that when the stamp duty 'holiday' ends that prices will fall. There are so many people chasing every property just now that it's a buyers' market. The trouble is, of course, that the longer he pays rent the more time it will take to save for a deposit - it is a bit of a vicious circle for their generation.