Just looked on dogs trust. No cockerpoos / cavapoos. One bichon, one Lhasa. Thirty Staffs, twenty one Staff crosses, 36 lurchers, 37 greyhounds.
You don't understand how the Dogs Trust works. Firstly, its main source of dogs are from other kennels - those operated by the RSPCA, dog wardens etc. Those kennels, if they get "popular dogs" can usually rehome them with ease, so the Trust takes dogs that deserve a second chance who would probably otherwise be put down. These tend not to be the popular dogs. Secondly, the Dogs Trust has many dogs that are not "on the record" - you can't see them. They may be in assessment or support, and working towards being rehomed. There is a reasonable chance that many lockdown adolescents (because they really aren't puppies) will be in this group, either because they are unknown quantities, or because they do have behavioural or other issues. It takes time for these dogs to filter into the general population. No dog will be rehomed unless the Trust are confident about its needs - they don't care about what people want, it is all about what the dogs need. Some dogs they take on may never, ever be rehomed - they are saved, but their issues are so deep seated that it is unlikely they will find a home that is suitable. Thirdly, a number of dogs will come from Canine Care scheme homes - homes where the owner has "willed" the dog to the Trust in the event of their own incapacitation or death. These are often dogs that have long-standing attachments to owners, usually come from "good homes", but may experience issues after the loss of their owner or, alternatively, be very easy to rehome quickly due to the nature of their previous home situation.
Whilst they do sometimes take in dogs from the public, this is actually very rare. You can't just rock up at a Centre and hand in your dog. In exceptional circumstances they may take a dog in - I have known of situations where people have tied their dog to the fence /gate post of a centre and abandoned it. But taking a dog from the public, outside of the Canine Care scheme, is not common. And they are not allowed to collect dogs or strays - that role is defined for the RSPCA, and they can't do it.
Behind the front page on the website there is work going on with hundreds more dogs.
That all said, if all someone wants is a cheap "designer dog", then aren't they as much to blame as everyone else. Lockdown puppies shouldn't be an excuse to pick up designer dogs at knockdown prices. If someone isn't willing to pay the price, whether that be for a puppy or an older dog, then why do they want the dog - because they would love a dog and really want to give a great dog a home, or because its a designer dog with status? Don't get me wrong - I hated seeing all the puppies appearing without any thought for their care or long term interests, and that prices have become astronomically silly all round. But is anyone wants to give a dog a home, there are plenty to go around. Just maybe not the exact breed you want, but isn't the dog the point, not the breed? They come in plenty of sizes even when not "designer".