How to get rid of excessive 'silly' mistakes?
First, stop thinking about "Tier A" or "Tier B" or whatever "tiers" -- it's another kind of a silly mistake. You have the whole Internet in front of you, so gaining knowledge anywhere is not an issue.

Next, after supervising many different students, I tend to believe that some people are inherently unable to be meticulous. We indeed should discipline ourselves, and simple tricks do help. The best one is to write, put aside your writing, do something else, re-read later carefully and fix.

However, in general I think the best advice is to focus on your strong parts. Yes, perhaps, you won't win a medal for attention to details. So try to train yourself to the best of your abilities, but think what kind of future job/career would not be too demanding in this regard. Exams are quite artificial situations, hopefully in "normal" life you won't have to do often something quick and correct while the clock is ticking.