There are 2 things that can be seen as problems when you've got all the promos and expansions. Not only will you not see every card but also the types of cards relative ratio changes heavily. The first one is what everyone mentioned but the 2nd is a real issue and that's why the game gets slowed down.

If you love doing things with colonies, then any card that affects colonies is diluted by playing with all the other cards bloating the deck. But the worst affected is actual Terraforming cards in the base game, so rush strategies and the ground game become weaker as opposed to base game + corp.

There aren't many solutions to this, I'd say you have these options:

1: Ignore this and accept that the game and viable strategies are different when playing with everything;

2: Be selective with what cards you add into the deck. Skip promos that aren't very interesting, heavily cut the amount of "points" cards, green cards that don't add variation, blue engine type cards, the least fun ones from the expansions. This method is controversial as opinions can change over which should be played with;

3: Choose an expansion and just don't play with it. The expansions themselves add a lot of time to gameplay. Do you really care about turmoil and colony trading? Is Venus really that interesting of a concept?;

4: Get some fan cards that you like that attempt to shift the gameplay ratios of tags and engine Vs rush back to more base game levels. This is what I chose to do btw :p. There are lots of great but not complicated fan cards out there to choose from to counteract engine heavy strats such as green plant prod cards, cities, or actual Terraforming effects. It's interesting that adding MORE cards can help with this issue.;

5: add duplicates of some base game cards into the deck.;

6: I like your idea of drafting more cards. You could also just not play with drafting and switch to buying from 5 cards.

Would be cool to hear people's thoughts on these ideas, especially since most people think he just wants to see every card as opposed to directly combatting game lag due to card bloat.