The police have nothing to do with it, they've had nothing to do with strays/found dogs for donkeys years now unless the dog is dangerously out of control of course.
Dogs have to do their 7 days in the council pound, unclaimed, to then legally be the property of initially the pound, and then whoever rehomes them from the pound.
Most pounds are also dog rescues, some are boarding kennels who then pass on to rescue - very few these days would PTS without attempting to rehome, one reason for that is, they'll get paid whichever way! They get paid from the LA for their stray contract, for the 7 day stay, and then when they rehome, they get paid an adoption fee there too.
The chances of a pound euthanising a PUPPY, are roughly the same as the chances of you finding rocking horse shit stuck to the hoof of your unicorn.
Your story is more than 'several years ago' isn't it - if the dog lived til 10 and died when exactly? Police did handle strays and yes a note saying a home was waiting may well be missed if the dogs 7 days were up on a different shift to the people you left the 'we'll take this one' note with. One reason why police do not deal with strays now, its a lot of work.
If you keep a found dog and do not notify the dog warden, then that dog is never legally yours should the real owner try to get them back. It is called 'theft by finding' and it does happen to pets and causes their real owners serious distress. Stories like yours that are outdated, dramatic and no longer relevant are WHY many people never see their missing dog again, because this myth persists.