Use the Nikon F. These cameras were made to be used and used hard. If you don't use it, the lubrication will dry out and you won't be able to use it.
The Nikon F2 was a more user-friendly camera than the F--and just as tough. That would be my first suggestion. I was not impressed with the F3 because many of the early models were unreliable. I hated the FM so much I never owned an FM2 or FM3. Nikkormats were good cameras and synched to 1/125, if I remember correctly.
But I would still recommend you use the F that you have.
I wore out several F's. They are great cameras but only sync at 1/60 if I remember rightly. I loved my F2 but in the end sold it and kept my Nikkormat FTN. Rock solid camera.
I use and really like F and F2.. If I could keep only one, it would be the F2.
The F2 is rock solid, fits the hands a little better than the F, which has German Contax inspired edges descended from the RF Nikons. F2 also has the 100% finder. Easier going with mirror lockup. Like the F, it has simple, direct controls, no battery dependency. Flexible. Lots of finders and accessories. The sync is a little faster on F2. The red mark is somplace between 60 and 125. Maybe an 80th? Some say that F2 was the pinnacle of Nikon's handfinished camera construction era. I think it may be true, but not by much. The F richly deserves its status as a classic.
I have owned Nikomats. Honestly, they are rock solid battle tank Nikons through and through as far as I can tell. I just do not like the cropped viewfinder. About 86% or some such dismal number as best as I recall. As has been noted, their sync is 1/125 with the vertically traveling copal square shutter. Best of all, they mount available and affordable Nikkors same as F, F2, F3, etc.