The deck tracker has some use(for future turns, e.g.), it however doesn’t help much with recognizing things under time pressure. Looking at all the current minions in the game without any stress will probably help a lot more with that, so i think Avondale’s advice is on point there.
Another thing that can help with your time management, is figuring out as early as possible how many tavern refreshes you’re going to use. If you take all the time in the world on the first selection, but then only have a couple seconds left for multiple refreshes+purchasing another minion, you’ll likely make bad decisions as you won’t have time. So, figuring your refreshes out early can make it much easier to dedicate the correct amount of time for every batch of minions that you’re offered that turn.
If you’re not aware of it yet, it can be useful to understand that playstyles/builds can usually be categorized as : Divine Shield, Deathrattle(baron rivendare specific), or as one of the 5 Tribes(Dragon etc). Most tavern and in-battle buffs are based on these. Knowing this will make it easier to realize what synergies/builds you’re likely going to be moving into, which can save time during your minion selection as you have goals.
If you’re looking for more than just tips on dealing with time, i’ve just started offering Battlegrounds teaching on this forum, so if you’re interested go check that out here Offering Battlegrounds help to beginners 