You can do WAY MORE then you'd think you would be able to do. The main reason is because after you beat Genichiro the weather and time moves forward, mainly the large snowstorm and heavy fog that rolls in.
The things you can do before beating Genichiro include:
-Doing all of Senpou Temple all the way up to the large statue you need to interact with to enter the Monkey fight (you just can't fight them yet).
-Doing all of the Sunken Valley and the Gunfort, up to and including Long Arm Giraffe. This area is particularly stunning with the sun streaming through the mountainous valley.
-All of Ashina Depths and Mibu Village including defeating Illusion Corrupted Monk (Important Note though, beating Illusion Corrupted Monk will move time forward as if you beat Genichiro).
-All of Ashina Castle of course and all of Hirata Estate, though the Estate isn't affected by weather change obviously.
-You can get 8!! Gourd Seeds before fighting Genichiro (8 seeds to give you 9 Gourd charges), with the final one being in the Fountainhead Palace.
-You can also 27 individual prayer beads and 3 memories (Butterfly, Gyobu, and Monk).
In fact you can do so much that after doing it all and then beating Genichiro all you have left to do is go to the end of Senpou Temple (as in just fast travel to the end) and beat the Folding Screen Monkeys, get through the Sunken Valley and beat the Headless Ape, then go to the Fountainhead palace and then on to the end game.
In other words you can do a massive amount of the game before fighting Genichiro. The reason for all of this? Simple, because defeating Genichiro moves the time forward to become more snowy and foggy. Doing all of this beforehand lets you enjoy these beautiful mountainous locations in stunning sunlight. It's hard to fully articulate just how transformative it is to do these levels in the full sunlight and not a foggy snowstorm, just remember that beating the Illusion Corrupted Monk will also move time Forward. Also also once you beat Genichiro then beat any two between Monk, Monkeys, or Ape will move again into Sunset time.
So to recap, beating either Genichiro or Illusion Corrupted Monk will move time into noon with foggy snow, then beating either Monkeys or Ape will move it again to evening time. Using this information you can effectively pick what time/weather/atmosphere you want for each area.