I don't know if it goes down as urban myth, slurs by people making products that compete with Neatsfoot, or if there is any truth in it anywhere. I'm sure all older horse people grew up with the sure and certain knowledge that it DOES rot stitching, I know I did. I have never had any rot though, so I got converted to the other side...I always have some on hand, I use it a lot reviving old tack, and sparingly on anything else.
What I have read is that back when the stitching was cotton, neatsfoot oil would rot it. (I still find old horse books now and then that warn not to oil near the stitching).
But newer tack has nylon stitching so neatsfoot will not rot nylon. I use neatsfoot on all my tack (because I am cheap I guess.....I really prefer Leather Therapy Conditioner or Skidmore's). But I have a whole gallon of neatsfoot and use it on everything! Never bothered my stitching. The only thing I ever ruined with neatsfoot was a cheapie pair of saddlebags that were made of suede. They were junk honestly. And the neatsfoot caused them to weaken and tear.
But all my western saddles and bridles get heavily neatsfooted at least twice a year.
I don't know how English riders manage to clean tack after every use. They spend as much time cleaning tack as they do riding! Maybe it's just that the saddles are smaller. I dunno. But I wouldn't be able to keep up if they needed cleaning that often.
I guess back to the OP's question, yes, once I am sure I am keeping a new saddle, I oil it. Because I would rather have it gently stretch than crack. And anywhere the leather is bent it could crack if not conditioned. So I do oil my new saddles. Sometimes more than once. It depends on how soft the leather is. Some of the newer saddles seem to come pre-conditioned. I don't know if Circle Y is one of them. It might be. I had a new Reinsman that came with soft leather. But anything else gets a heavy dose of oil. Anything pre-conditioned gets only a light coat of oil.