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My son is British as I arrived 5 years before his birth.
I worked I guess that helped. So sent my p60s no bother.
But as you say, not everyone will have kept paperwork.
And treaty rights are trickier if you didn't work, so if the other parent worked but is an arse and won't help you, it can cause hassle.
I also heard stories of British parents being unwilling to get the paperwork for the non British ex, result, kid can't get British citizenship application sorted.
Probably not - that requires the parent to have settled status, not simply be living here under FOM. Why would you have obtained settled status when the UK was part of the EU?