A Tele would actually be my first choice, but I've never actually owned one. Les Paul style guitars have very different playability because they're actually designed slightly different from most (I've been told this, I've never worked on a Les Paul). Where most guitars have the neck set flush or angled slightly up with respect to the body, LPs have the neck angled slightly down, with a raised bridge.

The input jack is a common failure on pretty much EVERY guitar. My Ovation (cheapest one they sell) is the only guitar I've owned that I've never had to repair the jack. Truss rod adjustment can definitely help get the playability there on your Tele, in my experience the entire Squier series needs (very slight) adjustment every few years.

Some more tips if you're going to give it a try: Pull back slightly on the headstock (bending the neck to tighten the strings) when making your adjustment, if you don't remove or loosen the strings. This will help remove tension on the inner mechanism, and make the adjuster easier to turn.I like to make adjustments with the guitar in tune, so you can easily check it with proper string tension. If the adjuster is stuck, work it back and forth until it turns easily. If the inner mechanism is corroded, this will loosen that mechanism up so you're less likely to damage it.

If you have all day to REALLY fine tune truss and bridge adjustment, maybe file a few frets that buzz too much, most Squiers can have feather-light playability at any fret, even with heavy guage strings. The problem with the Squiers is that they won't stay that way as long as the full price Fenders.