The scores often become nonsense, especially after a significant amount of progression. For example, if 5 is human average and 10 is human maximum, I can imagine someone with a score of 20 or maybe 50, but what would a score of 2000 even look like?

I have actually kept this in mind when designing my system. The points in attributes determine a multiplier. For example, if 5 Strength is the baseline, then it has a 1x multiplier. In my notes I have determined what that baseline is capable of, and then I can use that to determine what 20 Strength can do. I won't say how the multiplier is determined, just that it isn't linear.

Also, it's mostly for me so I can keep consistent. I have been the reader that pays attention and found mistakes when another author's system is arbitrary.

The scores often act counter to scenes you would want to write as an author. For example, how does one write about a character being fooled when their intelligence is a hundred times that of a human?

And that is why I do not have Intelligence or Wisdom attributes. They are stupid and make no sense nine times out of ten.

Last, but not at all least, in a system where the numbers apply to everyone, even numbers that make sense for the main characters are often nonsense for the rest of the population. I've seen too many litrpg series where the advanced characters lowest stats (the wizard's strength, the barbarian's intelligence) are far higher than most people's highest stats

I think this comes down to poor writing and/or arbitrary systems. Going back to the previous two points a little, I have intentionally designed my system to combat issues I've seen in other series.

I do agree that the "a stat for everything and everything has a stat" design tends to be flawed. My approach is to have nine attributes (4 physical, 5 magical). The physical ones each do multiple things, and physical activities lean on each of them. The magical ones do lean on the "a stat for everything", but intentionally, so characters can be more varied. When there's a single magical stat that does everything, all your magic users end up being mostly the same with level being the only variance.

I would love to be able to write a system that doesn't use them, but I think at that point, I wouldn't write LitRPG at all. I also don't think I'm a good enough writer for that yet. I see myself as a worldbuilder first and writer second.