"Make America Great" would be a fine, if bland slogan. Everybody wants that. It's not controversial, but also not distinctive in any way.
"Make America Greater than Ever" would be better. The implication being that we can do better, and be better. But they intentionally went with something else.
It seems to me, and as you have identified, that the "Again" is the key part of the phrase that drives the whole narrative. Here's the kicker: by nearly every objective measure, the country is safer, richer, more equal, and has a better overall quality of life today than at any point in history[1]. The only thing that has significantly declined over the last 40-50 years is "the percentage of total societal influence held by straight white men."
"Again" is the dog-whistle of misogyny, racism, and homophobia, wrapped up in the plausible deniability of nostalgia for an objectively worse time.
[1] There may be some room for disagreements here, primarily because of the first Trump administration and the pandemic years causing some backsliding, but this was especially true in 2016 when the slogan first really appeared, which is when it should be judged.
They actually took it from Reagan. But you could still maintain the same arguments, even though the times were a bit different. The "Again" is always a bit of a questionable part. "Make America Better" isn't as aggressive in the goal, but it's more honest and broad in who and what it's referring to. I'd rather have some solid ideas than a slogan, any day.