The short answer is that the distinction between the Renaissance and the Middle Ages is fairly widely recognized as false one, made up by those of the time and during the Enlightenment as a means of distinguishing the periods. A good answer addressing this by u/isatis_tinctoria is here https://www.reddit.com/r/AskHistorians/comments/8q6ycv/what_is_the_accepted_evaluation_of_why_europe/e0h4374/?utm_source=share&utm_medium=ios_app&utm_name=iossmf, but the entire thread is worth reading. (A few years ago I read (part of) Monsafi’s Renaissance Humanism, which confirms this, and what I was taught in high school and college; it’s worth looking at).

As you already seem to understand from your dismissal of the “Dark Ages” as useful term, the era from the fall of the Western Roman Empire to the post-plague era was not static, and the Renaissance had more common with that era than, say, the Enlightenment.

In other words, the term Renaissance is handy to refer to a few centuries, rather than saying 1350ish-early 1600s. But it is no longer considered a “rebirth.”

[I’m about to step out now into territory less familiar to me, so you may stop reading at this point]

Perhaps it is most useful in art history, but even then, I’m skeptical. There was absolutely a flourishing period of more “advanced” art during these centuries. But even in this context, it’s hard to establish an enormous cultural distinction between the eras; rather, it seems to be an evolution. Notably, Giotto is a traditional early herald of the Renaissance. But he trained under Cimabue, whose work is very evidently an inspiration for Giotto. Similarly, Dante could be considered early Renaissance. But with both we are diving as far back as the 13th century. Hence we again see that the term Renaissance, while useful as a means of referring to time period, it cannot very precisely refer to a definitively distinct era.