It's an honor that you agree with me. You have a LOT of knowledge about turfgrass.I agree with Freaky Fido.
Of course, Kentucky bluegrass is not originally from Kentucky--probably from Europe--brought over by early settlers.
Perennial ryegrass is fine--but it does not spread. It does not have rhizomes like bluegrass has.
A top-quality bluegrass is a good start. It germinates slowly and is slow for the first few weeks. Be sure it claims good disease resistance--to several lawn diseases. Denmark has some good seed companies. A blend intended for rugby, or football is a good choice. (Called soccer in the US).
Sports Turf Guide 32 This thumb rule was valid in the past with thatch-filled turf made-up of old varieties. Today with modern varieties and good thatch maipaper.ipapercms.dk
Be sure the seed is not too old. It might lose about 2 percent per month just in storage--so the germination could have been lower than expected.
Feel free to plant plenty of seed--no harm results (in my opinion) the strongest grass plants will survive.
However, if the grass becomes or remains thin for long periods--I suspect that shade could be the problem. Trim the trees--reduce the shade. Perennial rye does not do well in the shade, bluegrass is only fair in the shade. The top-rated chewings and fine fescue cultivars, like "Shadow" for instance would probably do fine.
Just to add to what you said about trimming trees. In my experience trimming the lower branches up to the 12 to 15 foot high range can make a big difference, letting a little light sneak in. That's if removing some or thinning them isn't a realistic option.