What's the reasoning for not waiting until the end of June?
I think it has a few factors.
The MLB draft is a bit unique given its size and timing. There are 40 rounds plus compensatory picks. And, with a few exceptions for special circumstances, drafted players aren't signing major league contracts as they are required to sign minor league contracts. Which is also a bit different as MLB drafts target filling a whole club organization from single A to MLB.
Also, with mid-summer as the key time for clubs to make final moves before the 31 July trade deadline, an early June draft and mid-July signing period allows for clubs to best make moves internally and leverage trade positions for either a post-season run or rebuild. And, it gets their new acquisitions on the field in their ball clubs before the season is over. Summer moves before the trade deadline are part of baseball culture as one of the most actively managed and fluid team rosters in sports.
It may be inconvenient for some college players, HS players, or even the amateur ball club players in the pool of prospects, but, honestly, MLB has no concern over that. Plus, I'd think college coaches would prefer this to the old mid-Aug signing periods when trying to anticipate what their needs are going into the Fall academic semesters.
Bottom line, imo, the current draft timing is the best fit for their business model. I don't see that changing. College baseball could always shorten the season or move the start date back.