I edit players all the time (real and drafted, with no consistency for when, and including both aesthetic things and actual attributes). It definitely does not stop them from progressing.
You mentioned you're still in your second year, which means your initial draftees haven't even had a full season yet. Little to no progression at that point is normal. Players often progress only a few points per year, even with elite potential.
An example: I drafted a 19-year-old SS who was a 54 overall with 90 potential. I edited him up to a 65. Now he's 22 years old, and he's a 72 overall. He may end up on my MLB roster in a couple years as a backup. I assume he'll be about a 76 natural overall, will get a bit of a morale boost for being in the majors, and play as about an 80.
At that point, he should start improving a bit more quickly until he reaches his potential.
(The same has proven true for guys I don't edit at all. I have years of draft classes and my full rosters saved in a spreadsheet, and progression is typically pretty slow.)
It seems progression in the minors is usually pretty slow unless a guy is just dominant, and many prospects come in as major projects, so they won't dominate.
My current MLB roster has a few examples of the dominant/quick progression types. Their potentials were about the same as that short stop (little lower), but their starting overall was much higher (low to mid 70s). They were good in the minors right away, they shot through the system, and now they're MLB stars.
That's really what you need to look for if you want quick progression. A guy with a decent potential rating and attributes close to big league caliber. Their hypothetical ceiling might be lower, but they're much more likely to reach it.
And yes, the players who don't enter the draft will be available in future drafts. I've never seen a guy disappear. They just get older (and, unfortunately, don't seem to get better). This is something that really should be fixed. The draft isn't remotely like the real draft, where every draft-eligible player is available and the decision to go pro or not comes after a team has picked you. Fully implementing that might be hard, so I can accept some differences, but I'd at least like to see every player available and the guys not picked just recycled back into the scouting pool. I've had some years where I don't even get to make my final pick because it runs out of players even though only a fraction of the guys I scouted were in the draft.
Also … players showing up in the scouting pool as teenagers and then not actually entering the draft until they're 25 needs to stop. Any halfway decent prospect will be drafted by the time he's 22 at the oldest. Anything older means he didn't go pro after high school, didn't leave his four-year school as a junior like most top prospects, didn't get drafted after his senior year either, and is still considered an MLB prospect. Just doesn't happen.