I am prolly not gonna make any friends with this post, nor is it gonna be a short one.
Lets try do this another way around. I have set up many of the mentioned decks, a girlfriend owned a TD160 IV, I owned and still own Linn (though severely Funked up) and owned a Planar 3 for a long time (b4 my first Linn TT).
First, unless you are buying your used belt drive turntable from someone who properly services them and spends a little money before putting them on the market besides cleaning them up, THEY WILL ALL NEED NEW BELTS. A belt is not worn when it falls of the pulley, by that time it is beyond worn and a disgrace to vinyl replay!!!
A shiny slippery belt (that may still be fairly snug) vs a fresh belt makes as big a difference to sound and presentation as cartridge upgrade. The decks will sound lifeless and uninteresting
Iow, imo, you have not yet heard the capability of any of these decks! Keep that in mind when comparing to new!
Belt prices for real replacements, not equivalents from the respective importers at last check: Thorens R300, Rega R350 and Linn R550 (No wonder thus that most used decks have shagged out belts)
The Rega imo is the worst buy, the price is too high, the market price for a Planar 3/RB300 is between R2500 and R3500 in good to mint condition without motor upgrades etc. I believe this sample even has a dodgy lid. Broken lid makes one wonder in what other ways the deck may have been mis-treated? New lid is R1K+ I think
Besides that, I would feel the Rega can be a little stark (lightish) in the bass regions. Poorly recorded thin sounding LPs may not be so much fun to listen too (Punk?). This seems a trait with lightweight unsuspended decks, with a few exceptions.
Set up is simple, though even Rega occasionally needs motor angle adjustment...
I believe both the Linn and Thorens are decent buys (and as fussy about set up as any loosely suspended deck) but not without one or two things to consider.
The Thorens is a good deck, the particular model on sale is apparently a "UK importer tweaked" special edition. This means that the standard stuff people may do to Thorens decks have allready been done. The Linn Basik arm on it is OK for MM cartridges but left far behind by an RB300. The deck has more influence then the arm on the overall sound though and some who do not understand Rega arms (on a suspended deck) would say the Basik is better matched then an RB300 is on either Linn or Thorens... IMO not true if you know how to treat a Rega arm. Thorens decks tend to sound a little more laid back in standard guise then the Rega or Linn. Together with a PM17 which though a finely balanced sounding amplifier, is not an adrenaline pumper. Keep that in mind while playing Punk. Overall, a well priced buy though
The Linn: An ikon, a place where real stuff starts for many (spelling mistake deliberate).
Upgrade options are endless, from Linn and others. Some do not really understand the deck and their mods can stuff it up imo but if done well it competes with anything out there. German mags (less influenced by fashion) still consider the Majik and the SE LP12 "Spitzenklasse" along some seriously dear other contenders.
Because they are an ikon product, reselling is typically easier then other decks so can be a safe way of dipping the toes.
The Rega arm can be made to swing on the LP12, it just should not be torqued up with a "Monkey wrench" like most Linnies will do. A little more then finger tight, a serrated washer below does wonders.
Set up on any suspended deck is a little more involved then an unsuspended deck. However, with "black springs" and nylock or "castle" nuts, a correctly set up Linn (or Thorens) is quite stable.
With regard to 45 RPM, Linn offer an adaptor but it messes with belt tension, lopsides the suspension, is a pain to fit and is just in general a great reason to upgrade a Linn powersupply to 45 rpm capable ones

I have just brought in the first batch of Hercules powersupplies, a drop in replacement upgrade on Valhalla that plays 45 rpm as well.
I can also modify Valhalla supplies to have a little external black box with a 33/45 toggle switch from a fly lead. This is quite cost effective.
The downside to the LP12, mmm, other then expense associated with 45rpm, just one sonic one. The cartridge represents a fair amount of the value of this sale. I am not convinced, fine as a Rondo Red is, that it is the best choice for rock. When that one is worn out, I would prolly choose something else. Second, that dark smoked lid is not exactly hip and not cheap to replace either.
An RPM5 or P3 imo will not come near the Thorens, never mind the Linn.
The Linn IMO offers best value, even though it is not the cheapest here.
WRT supports, any turntable, except those with their own hydraulic/air suspension or weigh as much as your cars engine (that has sonic drawbacks too....)