I maxed out the research level of the first stage, but I didn't notice any difference. Does anything change?
Maxing out research doesn't result in any course changes. Similar to you I haven't noticed any changes on the few courses I've maxed out, but the way you can confirm this is by manually selecting your research level.
For Park (Day), I've maxed out research. If I manually select the research level by pressing X, I see a menu showing research levels 1 through 3:
Of course it's entirely possible the research menu only shows levels below your max, so I decided to rule this out by viewing the research menu for a course I haven't maxed out yet. For this test I chose the Illumina Spot on Belusylva, on which I've only unlocked research level 2. If I open the research menu here, I only see two research levels:
The combination of all of the above is fairly strong confirmation that "MAX Lv." doesn't actually change anything in the course itself.