Just yesterday, I spent several frustrating hours trying to find t-shirts that fit my requirements, which are somewhat different than yours.
I want a t-shirt to be as thin a material as possible. The whole point of a t-shirt is to either wear it as an undershirt or standalone in warm weather. Thick material that is more durable is antithetical to either use.
"Fits well" is stupid hard to find. The vast majority of t-shirts are boxy with straight lines that have zero form fit. The sleeves come down so low and have such a large circumference that they cover any muscle you might have. They are also long enough to cover most of your ass, which is overkill for tucking and looks super dorky when untucked. Of course all if this is relative to your dimensions. I am 5'11" and reasonably fit. Even if you are not fit, the standard cut looks horrible.
Another aspect of looks is color. There are always the boring generics: white/black/mottled light grey and sometimes navy. For standalone use, there are some few outfits that these colors work for. As an undershirt, these colors work more often, but I have found that there can be better color matches if the undershirt is visible (and especially if you are doing the west coast thing with an unstructured sport coat over a t-shirt).
That covers much of the (lame as hell) t-shirt market, but of course there are some standout brands with better features. If you asked me 2 years ago, I would have said True Classic as they have a more tailored cut, reasonable length and arm circumference and better selection of colors. The material is 60/40 cotton/poly and not too thick (but not very thin either). Recently, they changed the length (added 3 inches!) and ditched many of the cool colors, which I think is due to this inane fashion ideology of change it up annually and these are this year's "cool" colors.
The ultimate shirt is one that is tailored to your spec and fit. There are some online options for that, but they all cost more than buying a generic shirt and spending ~$30 to have it tailored (or DIY: https://www.youtube.com/watch?v=XeWUPBeL5XM).
If you are in the EU, Asos has some branded shirts that are a step up and reasonably priced.
https://freshcleantees.com/ is a True Classic dupe, but I haven't tried them.
I ended up going with TC, despite the lack of colors I wanted because it is pretty easy to shorten the length.