The problem of matchmaking is actually quite a complicated one. Here are some thoughts on it:
1. Familiar cards is exactly what you're looking for. This feature allows you to say which cards you already know, and how many unknowns you want to allow in your kingdom.
2. The problem with the matchmaker that respects familiar cards ("practice games") is mostly that it isn't popular enough. It takes quite a bit longer to find a match there. It's less people, and it's harder to find a match because not every pair is valid (you may want to play with base+intrigue, but maybe someone else wants prosperity+hinterlands).
3. People are a bit spoiled by how fast rated matchmaking now is. I had to wait 10-15 minutes myself on average on any of the previous implementations (iso, goko, MF).
4. @JW the problem can't be solved by making the games part of the rated matching. Yes rated matching is more popular, but that is because more people want to play with all the cards they can get their hands on. Making "practice game matching" somehow part of "rated matching with special options" would just compartmentalize the player pool in a way that is less explicit, but it wouldn't magically allow a match between a player that wants to play with everything and a player that wants to play with 1 or 2 sets.
5. Alternatively you can skip the matchmaker altogether and find a match yourself. It would be nice if the client would make it easier to talk to friends, it will one day. Hanging around here
also works. Creating a table and just waiting could also be improved, by making it easier for other people to spot your table. One day it will be.
6. At some point we'll add Campaigns as an alternative way to learn the new cards.
7. We could add some concept of "pretend you don't have subscribed to certain expansions you actually subscribed to". I doubt it would be very popular, but maybe I'm wrong there.
8. There won't be any cheaper subscription models. That's just an economical argument - Silver and Gold are already so cheap that almost nobody will reject it because of the amount. If people don't subscribe it's because they just don't want to pay for various reasons. And people playing with less cards aren't actually cheaper to support for me then people playing with all cards (assuming those cards get implemented anyway).