Is there an 'Audiophile' Reel-to-Reel Machine for Under $1,000?

I haven't heard of one. Maybe $2k if you are very lucky, and you may still have to service it soon enough.

I will also mention Technics.

Yeah, you could get a good sounding deck for $1k but it may not be good enough. That's the main reason why I don't have reel-to-reel. I saw Otari MX-5050BII-2 both on ebay and on reverb for close to $3k in seemingly good or better condition. That's where I would start. Studer 810/807 is two/three times more.

Revox in good condition will be over $1k.

I think, we made it sound very complicated to the OP. But, yes, he must decide if he needs two or four track deck and if he needs 15ips speed.  Master tape dubs can be less than $400 but still very expensive. For his old recorded tapes he may need 3.75ips speed, we don't know. For less expensive deck than Otari I would probably go after either Pioneer or Revox.