For flights within Europe, are there times of day for which the airplane is emptiest compared to other hours? I was thinking, for example, a flight at 5 in the morning probably is the least popular and so the plane most likely is not filled the same way a can of sardines is!
A crude and inexact, but quick and easy, heuristic is to look at flight prices. All things being equal, for direct flights between A and B, the emptier (less popular) ones will be cheaper and the more popular ones will be more expensive. This is because airfare pricing works on price "buckets" (fare classes) that fill up as the airplane fills up.
Note that it's important to limit your search to nonstop flights only, because otherwise you'll get connecting flights in the mix that cannot easily be compared.
https://www.rd.com/article/secrets-to-avoid-flying-on-a-crowded-plane/ claims that flights in the middle of the day and on Tuesday, Wednesday, and Saturday tend to be less crowded.
If you can be flexible on what day you fly, you’ll have a better chance of finding less crowded flights. Flights on Tuesday, Wednesday, and Saturday tend to be less crowded (and cheaper!) than flights on Mondays and Fridays, says Scott Keyes, the founder and chief flight expert of Scott’s Cheap Flights. If you want to avoid delays, this is the best time to fly.
The most popular time of day to fly is early morning and early evening; this is also when flights can be expected to be most crowded, says Keyes. When you’re choosing flights, pick a nonpeak time (these usually correspond to when flights are cheapest as well); flights in the middle of the day tend to be less crowded than morning or evening flights. Or try middle of the night flights; not only are these often less crowded, but passengers will often be sleeping, which means fewer people walking in the aisles or conversing with each other.
Note that this is just a very vague approximation. The actual airport and route are likely to matter to some extent.
To estimate the occupancy of a given flight: https://thepointsguy.com/guide/how-full-is-my-flight/
Early and late flights are full of business travellers, especially on Tuesdays and Thursdays as people often work Tue-Thu on remote projects (why is another topic, but I've done that schedule myself and the flights at 6 am and 9 pm were quite crowded).
The most empty plane I have ever seen had me and two other people on it, plus the crew. That was an early Sunday morning flight. Sunday morning cuts out the two largest crowds - business people and tourists. Business people will travel during the week, and tourists would start their holiday on a Fri, Sat or Mon more likely than a Sunday, and are more likely to end their holiday with a late-in-the-day flight than an early morning one (even the red-eye flights don't come in a 6 am).
There are times when there are fewer people flying ... but that just means there are less flights, not emptier planes. Airlines need flights to be mostly full just to turn a profit, and make more money the more full the flight is. They've gotten pretty good at scheduling / pricing flights to ensure the flights are almost entirely full. If there were a time when flights are emptier, airlines would drop the prices for those flights to try to get more people to fill them, and in the future reduce capacity then by having less flights or smaller planes.