While I recognize that a lot of people see the Gillette (and similar) ads as not a big deal, here's my problems with them:
They presume me (all men) to have already committed a transgression. Not particular men - all men. Harvey Weinstein does something and I'm guilty by gender association. I need to "do better". I need to "be better".
Maybe it's my own social surroundings, but whenever I hear or have heard "be better" being used, I take it as a negative statement: "You're a bad person and you need to be better". It assumes bad behavior (or lack of good behavior) and scolds us to "do a better job". Statements like Gillette is making assume this guilt by association. Some men did something bad, so all men have to face some consequences and all men have to apologize.
Melania Trump has the "be best" campaign that everyone makes fun of. While the phrase is a bit clunky, I read it as "be your best". This is (again, based on my own upbringing and social surroundings) a positive phrase. It does not assume I have done something wrong that I need to atone for. It assumes that I'm already trying to do good (or at least doesn't assume I need some kind of correction), and just tells me to be the best I can be every day.
Be honorable. Tell the truth. Be your best. These are things that "toxic masculinity" has been teaching us for literally thousands of years. Gillette could have easily gone with "Be your best" or "Be the best man you can be" and left it at that. They didn't need to drop "me too" and other "bad behavior" references into the ad. Of course I guess that would have contributed to the "toxic masculinity".
I think zahc hit the nail on the head in his post above regarding who/what drives much of this nonsense.