Hope I don't get too much grief, but I politely and firmly disagree with Ellen - recent Squiers have a variety of neck thickness ranging from thin to medium. While you won't find a 'club' like a 50's Les Paul tribute, things have changed esp. since 2012-13.
OK, Teles. Few thoughts ...
First rule of FMIC guitars: QC?
We don't need no stinking QC!!! :
aughter:: You
always play them all, because there's a chance (albeit small) that an Affinity might have the best neck of the lot. The price point where this changes is above American Standard and Elitist. Play all the ones in the store. Do the odds change with the price tag? Yup. And so do the metal-based parts (electronics and hardware). BUT ... necks are truly all over the place. Which brings up ....
Second: ... at the range in which you're shopping, the least easily changed things are the fretboard and looks. So, shop for neck feel and looks. Lots of people change pups, bridge, tuners, switches, etc. Not a biggie with a Tele. You asked how a V neck is different .... how do I explain this? I LOVE shallow-V neck profiles, but you might not. Find one. Hold it. It's the only way.
Third: having an 'issue' with the Squier name really is - how to say this nicely? - quite outdated. At this point, nearly embarrassingly so. Things changed dramatically in the late 00's, and have been changing since. I have a couple Squiers I would not trade for their Fender USA counterparts. However ....
Fourth: .... if you really dislike having a lot of glossy finish on a neck, you
can cross a bunch of Squiers off your list. I think this is done to try to cover up their using greener wood. Then again, I have had Squiers out on stands in a desert environment for nearly 10 years and they're holding up fine.
Fifth: I don't think there's a huge intonation issue with the three-barrel bridges, unless you play a lot of chording up the neck. Anyhow, Tele hardware is plentiful and easily swapped.
My teles:
'73 US Custom
'94 MIM two-piece (!!!!!) body
'07 Squier Affinity
'12 Squier CV 50's RI (the Keith Richards blonde thing)
'11 MIC Modern Player
'02 MIM CP thinline
All are enjoyable to play - and in no case do I ever think "for the money" when I am playing them. They're Teles. Teles are either 'good Teles,' or 'not good Teles.' You meet good Teles in person.