Its definitely better to be a little earlier than later. Couple factors are temps like you said, and also increasing shade as the sun angle lowers. Last year I had to do an emergency seeding of an area like 20 x 20 that was an old garden bed that we were relocating. It was sometime late in mid/late Sept I want to say I did it which is way late in this area, which i knew but tried it anyway. It all came up ok, but never grew much as soil temps got chilly into OCT. The biggest factor in not being totally successful was that area went from getting 10 hours of sun to 2 hours by OCT, so it had shade and soil temps dropping too fast.. If had started in AUG prob no issues.
Im starting an overseed right now in a small section to give it time to establish.