I wonder if lose-track could never have existed in the first place.
First of all, you have the cases where a card is covered. The problem is that you have stuff like Warrior finding a Tunnel and not being able to trash it because it gets covered by a Gold. I'm pretty sure almost 100% of people who play IRL plays that wrong, because nobody thinks about that rule applying then. Arguably that also goes for all cases with triggered gains on when-gain (or triggered gains on when-trash resulting from when-gain). Almost everybody not intimately familiar with lose-track (or having played a lot online very observantly), wouldn't think that you can't move the card.
Secondly, lose-track is the reason cards played from trash don't jump out*, which was relevant from day 1. But in most cases people don't need to know the rule, because that's the way they play it intuitively - the opposite of Warrior/Tunnel. Of course you need a rule, but it could just be that cards played from trash stay there.
The third type is when the card is actually lost track of for the players. I can think of Inn shuffling in a card. In this case it's obvious that it's lost track of; you hardly need a rule, but it could be a rule about shuffling a card.
The fourth type is cases where a card moved. For instance, abilities moving a card twice: Counterfeit and Procession don't trash a card that moved in the meantime; Vassal doesn't put a Faithful Hound in play after it was set aside; Summon or Replace don't move a gained card that was moved by Watchtower/Royal Seal/Tracker/Travelleing Fair. Or multiple abilities triggered by a card moving: Alchemist, Scheme, Hermit, Prince and Faithful Hound all trigger on discard, so then just one of them can move the card; and we also have Fortress + Possession.
My question is whether lose-track is needed to regulate this type of interaction. Why not just allow that everything happens? All abilities move the card. Well, one weirdness with Watchtower is that it would let you move a card from trash without even gaining it. We could revise the rule about playing from trash to cover both cases: "A card can't move from trash unless an ability lets you can gain it from there." Another weirdness would be that Hound would be put in your hand at end of turn but still be played by Prince the next turn, but I guess that would be okay. (It would be an optional and dumb move.) (Edit: Not a real case, since a Princed Hound can't use its ability.) I can't think of other problems right now.
The fifth type is cards specifically designed to exploit lose-track, like Madman, Prince, Wish. These can't be successfully Throned because of lose-track. A rule about moving cards from trash doesn't help, because these cards move from somewhere else. This type might be a bigger problem. The problem is purely with TR variants. A fix could be to make all TR variants like Royal Carriage: "You may play a card from your hand, then if it's still in play, replay it". (It would mean that Madman and Wish doesn't even give +Actions on second play, but that doesn't really matter.**)
Conclusion: Lose-track is complicated, so I probably haven't covered all cases, but if I designed Dominion from scratch, I might try to do it this way:
Rule 1: A card can't move from trash unless an ability lets you can gain it from there.
Rule 2: A card that is shuffled in a face-down deck can't be moved.
Change Throne Room variants to: "You may play a card from your hand, then if it's still in play, replay it".
The point of these two rules is that they are intuitive. Someone might wonder about moving cards from trash, but most people wouldn't assume that you can without checking the rules at least.
*except when an ability tells you to play a card from the trash: only Necromancer, which specifically prohibits the card from moving.
**Late edit: It would of course, as Donald has pointed out, also mean that things like TR+Feast doesn't work.