Most of them have undisclosed behavioural issues and no onward care is offered so your totally alone in managing the (awful) issues they have.
The rest are either stolen (as happened to my colleague who's dog was stolen out of her garden and later found on one of these websites), bred for the sole purpose of being transported to the UK, riddled with all manner of infection or diseases, or completely unsuitable as family pets given their generic make-up.
Half these rescues, including one suggested above, rescue, are a joke. Look at the background they give on the dogs - literally 3-4 bullet points. No actual information at all. No thorough rescue, that takes animal welfare and human safety seriously would ever provide such a small amount of information. When my breed specific one does it we have pages, and I mean pages, of information. Like Spaniel Aid does. Not four sodding sentences.
And importing an animal when there are thousands here that need homes, says a lot about the potential owner. It's about 'feeling about doing the right thing' rather than doing the right thing.
Sorry but if you are fit and active, presumably home all day if you're retired and have 6ft fences then there is another reason why you are being turned away. There are very few reasons rescues would turn down people such as you have described yourself, so there must be a glaring red flag somewhere.