It all depends where you go. The only really appallingly busy trail I can think of on Bank Holidays and the like is the paved stretch between Mam Tor and Hollins Cross plus the obvious converted railway trails and roadside tracks – Ladybower for example – which get understandably rammed with families. Get more than about ten minutes from the carparks and it all quietens down nicely. Maybe you’re choosing the wrong trails.
I’m not saying there aren’t walkers on other tracks, but as above, slow down, be considerate, smile and carry on. I can’t remember having a genuine issue with a dog, bar having to occasionally slow down and equestrians seem generally polite and considerate – mostly I just stop and wait for them to go past if they’re oncoming or say hello and ask if it’s okay to pass if they’re going in the same direction.
I live on the Manchester side of the Peak and honestly, some of my best days have been all-day meanders from door on a BH weekend, out into the central Peak then back. Sure, you meet the odd walker, but I’ve never found it a big deal.