I adopted an older dog in 2009, having found her via The Oldies Club which aggregates info from shelters and rehoming organisations across the UK:
www.oldies.org.uk/
I adopted another older dog in 2019, having found her via DogsBlog, which also aggregates info from shelters and rehoming organisations across the UK:
www.dogsblog.com/
Both sites allow you to filter information according to numerous specifications like "Good with kids" (Oldies), "Not Cat Tested" (DogsBlog) - there are loads of these sorts of criteria on both sites.
This might get over some of the problems reported by PP, of organisations deciding whether or not you are suitable as adopters: you choose the criteria that match what you have to offer and then you only get to see a list of dogs that are a good match.
I also looked at other sites that advertised dogs for rehoming but I found those ones the most useful starting points because of the ability to filter by specific criteria.
My main reason for posting though is that I noticed a huge change in the dogs available across the UK between 2009 and 2019.
In 2009 it was very rare to see mention of a dog who had been born abroad.
In 2019 there were masses of dogs for rehoming had been imported or were still abroad and were looking for people to fund their care abroad, importation and rehoming in the UK.
Those that had already been imported were a mix of "new imports" and dogs that had been handed in to shelters, or abandoned, by the people who had originally adopted them from abroad.
This is an interesting article:
Rescuing dogs from abroad – does it make sense?
It concludes:
"SO, does it make sense to import rescue dogs from foreign fields or streets? Probably not.
Do these dogs impede domestic unwanted from finding new homes? Probably not.
Are the oft stated risks and problems caused by these dogs over-emphasized? Probably yes.
It is undoubtedly a messy situation which causes considerable angst amongst various animal charities and the veterinary profession, but I’m sure you will get no argument from those individuals who have been saved from the wretched conditions they have been kept in certain countries and given the chance of a life worth living."
animalrightsandwrongs.uk/2018/09/rescuing-dogs-from-abroad/
If people are uncertain about adopting an imported dog (rather than being for or against) perhaps they could consider dogs who were imported but have since been returned to a rehoming organisation?
The good news is that Oldies Club has abandoned its "Overlooked Oldies" list because it is not needed at the moment:
www.oldies.org.uk/2020/we-arent-using-the-overlooked-page-at-the-moment
There are plenty of other dogs listed there though, including dogs who have been imported:
www.oldies.org.uk/category/adopt-an-oldie
There are also plenty of older dogs on DogsBlog, including dogs who have been imported:
www.dogsblog.com/category/age/6-years/