There are hundreds of tele variations.
I owned a half dozen and ended up with a cherry sunburst Fender USA Deluxe used or 999 at GC, and Sam Ash gave me a sweet deal on a used G&L ASAT Deluxe. Tele bodies vary wildly. The G&L is a hunk of mahogany capped with matched flamed maple top, Birdseye maple neck, rosewood plank, Duncan buckers, 59 & JB. It is a tele shaped 80's LP. The Fender is maple neck plank, alder body. The sound is in there.
A tele fender authentic type likes a different amp than a Humbucker guitar IMO. One of the best Tele amps that is somewhat cheap is a Zinky Velvet. Old Fenders are not cheap IMO. Bruce Zinky designed the little beast and it wails.... YT search it. Amazing little amp. High gain side is the weak spot though. Dr. Z is another tele amp. I use a Bogner New Yorker with mine and love it. Lots of people use high gain amps, but I think you miss part of what makes a tele great. Jimmy Page worked much magic with them though.
A good compressor is part of the magic. Some swear by the Barber Tone Press, parallel circuit with a mix level, Keeley, or Wampler. I have a Barber and Wampler. Wampler is in use.
The fender tex mex tele with the Strat pup in the middle was a great cheap tele. I think 599 new. A buddy has a P90 tele, but that's in a hot rod class to me. A used tex mex 3 pup is a solid bang for he buck. The Strat middle pup in a tele is money.
Nash has some sweet ones. They use the Lollar bridge pup. Those are just slightly overwound and don't mess up the tele vibe. I have the Strat special set in my old Strat. Good close to vintage pups.
I think the tele is the most under rated guitar out there. The P90 version is the toss up with the LP Junior for that P90 type of guitar I want to get. Steve Morse had a Franken-Tele back in the Dixie Dregs days.
The Anderson tele looks nice. Suhr as well. Fat coin. I have played a few custom teles, the fit and feel was awesome, but the basic tone is in the 600$ tele too. Seems one of the guitars the mex versions are pretty stellar.
Good luck on the quest. So many good players on the cheap to be had.
YMMV