Along with pets, I also have children who spill juice and a honey who has dirty boots and won't leave them at the door.
You need to teach them the "Leave It!" command!
I don't like carpet. No one will ever convince me that you can get it truly clean.
I love carpet but agree it probably never gets truly clean. How can you get all the way to the padding and floor? Our new carpet (when we built 12 years ago) had a pretty nice 'scotch coating' on it. Anything that spilled, just puddled on top! But shampooing eventually erases that. I don't know if a scotch coat in a spray bottle is the same but it might be worth a try. I just forget to try it.

I just do my best with keeping it clean and perhaps getting new every so often. Our bedroom is due. It gets the most traffic. Our sunroom is right off of it and I'm certain the is an amount of dirt/dust/smoke from our wood burner in there. Plus dogs sleeping on the floor next to my bed.
We take shoes off at the door. I was raised like that and its just ingrained. lol Do you realize how much dirt/grime/oil (from parking lots!) that get on your shoes? Off at the door, I say!
I will say that when I do it myself, I use vinegar when I'm done cleaning. Just go over the carpet with a vinegar and water mix. Stinks while you are doing it but vinegar is a great deodorizer and everything smells fresh when it dries.
I use vinegar in a lot of things, including the shampooer! And you are right, it may smell while you are doing it (I kinda like that smell

) but its gone afterwards.
Professional extraction (steam) cleaning is the only way to go. I was in the business for many years, and have tried them all - extraction is the best.
I believe this too. I've heard many folks say that the leftover soap in the carpet actually attracts more dirt. Makes sense. And it's one of the reasons why I bought the Hoover that also has a rinse only button. I will use very little soap and then I'll go over it with the rinse again too.
My dream floor for dogs would be honed concrete,
Tile is cool,but hard on the back if you go bare foot.
I would never go concrete. Our house has a LOT of tile and I think, like concrete, it's hard on the feet. We've always had ceramic tile in our foyers, kitchen and baths. It's beautiful and easy to clean, but the older I get the more I think it is really hard on the feet. Not so much the back, the feet. I end up with throw rugs down everywhere! In fact, I'm going to get one of those thick, rubbery things for in front of the sink. Even if I put it UNDER the throw rug! lol