I've found that everyone has their own special way of conditioning and cleaning leather. Some people use nothing but special balsams, some people swear by nothing but saddlesoap and neatsfoot. And others prefer extra virgin olive oil. The point is: So long as you clean it and condition it regularly, it's down to a personal preference. (But until you're well versed on them all, you may want to stick with products you know are meant for saddles/bridles.)
Neatsfoot is more than fine. Many people swear by it. I would also get some saddle soap so that you can clean your tack and lightly condition it in between oilings.
I do saddle soap/Lexol Leather Cleaner every other lesson-ish, and also use a leather balsam, like the Passier Lederbalsam* for light/medium conditionings through out the year. Twice a year (once if it looks good) I will break down the saddle and give it a deep conditioning, using neatsfoot/mink oil or Lexol Leather Dressing. (Which is neatsfoot based, but won't darken the leather.)
You can play around with your routine, see what you like and see what you don't.
* Mine's similar, but not the same. In fact, mine was meant for use on Doc Marten boots!