Tolkien himself equated Middle-earth's weapons and armour to that depicted on the Bayeux Tapestry, so let's take 1066 as our reference. Were Britain and France comparably desolate and depopulated?
This isn't an exact answer, but I'll link you to a previous post I made on density in the middle ages in Europe here: https://www.reddit.com/r/AskHistorians/comments/751y50/my_history_teacher_once_mentioned_that_in_the/
In essence, for most of history back through to the start of the Iron Age Europe was pretty significantly settled. That doesn't mean that you couldn't wander around miles away from anyone else for a while, but you would have a hard time doing so for more than a few days. In general most of Europe was at most a single day's walk from a village or town and generally closer than that to some sign of civilization such as a house, a farmed field, a livestock pasture, a road, or such-like. And this was especially true of France and Britain during the middle ages. The standard middle ages stand-in fantasy setting where there are vast swathes of absolutely wild territory (uninhabited forests, grass land, etc.) is precisely fantastical. It's as unrealistic as elves or dragons. And as I mention in the other post it's something that by Tolkien's time hadn't existed in Europe since about three millenia earlier.