I am sure there will be some who feel differently and or it is done differently in their neck of the horse world. But I have been on LOTS of farms over the years, have puts up LOTS of fencing, miles and miles of it. First going back 50 years, when I was a kid.
The standard used by the majority of horse farms be it a high end TB or a small hobby farm are 8 foot posts sunk 3’ to 3’-6 depending on the desired height of the top board. The majority of TB fencing IME is 4 foot. Our weanling paddocks are 4’-6. A breeding stallion paddock is usually 5+’.
The horse fencing standard posts in my heavily horse populated area are 4x6 pressure treated for board and for wire. Though a lot of farms that use V-mesh wire also use round posts because they can be less expensive due to less “milling”, Half rounds are also used/popular. Rounds and half rounds are not always available in a lot of areas.
Before pressure treated became the standard creosoted post were the standard. But creosote has been banned in a lot of states for environmental reasons. It is also nasty stuff to work with, burns skin easily. Pretty hard to avoid unless one is clothed head to toe when setting post and boards.
When untreated posts were used what Casper suggested was pretty much SOP. Added a lot of extra labor and mess. It is unnecessary if using pressure treated posts. Throwing gravel in the bottom isn’t needed either. Nor is cutting the tops at an angle to shed water. I pulled some pressure treated post that have been in the ground going on 18 years the other day. There were in pretty much perfect condition. Totally reusable. We get a lot of rain around here also. And cold winters, frost line is 30-36 inches.
IMO which is based on lots of experience and based on your description 8’ posts sunk 3’+ feet will do job. If the post can’t be pounded which is the BEST way to set posts. I use a 6’ arguer for 4x6 posts, spray a bright paint line on it for a 3’ “gauge” and go down the marked fence line holes and drill away. Making sure the auger is “plumb” when drilling.
I have my assistant place and hold the post in the hole and press it in with the bucket on my tractor. They go in nice and tight with minimum back filling needed, little to no tamping. I generally mark the 3 foot depth on the post with a “fat” black marker because some dirt usually falls back into the hole when pulling the auger out. I pound the post to the line with my bucket.
Any posts that “stand proud” after nailing the top board are cut off with a chain saw. I only cement posts that go in shallow, 2± maybe 2-6 because of hitting rocks that are too time consuming to get out and or can leave a very large diameter hole.