TL;DR: No. Class C airspace comes with a specific set of requirements for pilots and a specific set of requirements for ATC. TFRs have "squawk and talk" requirements for pilots, but that does not mean ATC has Class C requirements.
With all TFRs, it is imperative to read the actual text of the TFR itself and not make assumptions based on what similar TFRs may have required in the past. Let's examine this VIP (POTUS) TFR, which is FDC NOTAM 4/4691. Generally POTUS TFRs will always be the same, but just keep that in mind.
The overarching restriction is "No pilots may operate an aircraft in the areas covered by this NOTAM (except as described)." We knew that, but it's important to read this anyway; some TFRs only apply to certain types of aircraft. It makes you sound unprepared when you request permission to fly your Skyhawk into a "No UAS" TFR.
Then they list exceptions to, and more details about, the blanket restriction. Note that it is possible for ATC to authorize other exceptions "in consultation with the Air Traffic Security Coordinator via the Domestic Events Network." In your linked question the controller probably did not do this coordination and so it was probably an error on their part. But in theory ATC can authorize any deviation from the blanket restriction. In theory.
Paragraph A is for the "inner core" of the TFR and basically says you aren't flying your private Skyhawk into or out of there, no way no how.
Paragraph B is for the "outer ring" of the TFR and now things become relevant. Among other restrictions (no loitering, etc) it says:
ACFT MUST BE SQUAWKING THE DISCRETE CODE PRIOR TO DEP AND AT ALL TIMES WHILE IN THE TFR AND MUST REMAIN IN TWO-WAY RADIO COM WITH ATC.
This means exactly what it says: You must be on ATC's frequency as long as you're physically able to be (line-of-sight to ATC's radio antenna). If you feel that you need to be on CTAF, then in my mind you have two options: Make sure your aircraft has two operating radios, so you can transmit and listen on CTAF while still monitoring ATC... or wait for the TFR to end. I'm not a lawyer but that's how I would interpret it. Remember though that CTAF calls aren't mandatory—just highly highly encouraged.
In contrast, for Class C there is more to the airspace than just "squawk and talk." (In fact the "squawk" requirement is also different between TFRs and Class C airspace. The TFR requires a discrete code, while Class C only requires an "operable transponder" responding on the ATC-specified code, which may be the non-discrete 1200 code.)
Within Class C airspace ATC will separate IFR-to-VFR traffic, and this service is not part of the TFR. Within Class C airspace ATC will sequence VFR traffic to the primary airport, and this service is also not part of the TFR; note that even in actual Charlie airspace you can have non-towered satellite airports within the surface area, and ATC will not sequence to those non-towered airports.
For a TFR, you will contact ATC on the ground to get your discrete squawk code and your release from the airport. As always, ATC will never issue any runway clearances (takeoff, landing, option) for a non-towered airport.
As for pattern work: Again, read the TFR. Paragraph C lists the many types of operations which are specifically not authorized within the TFR:
FLT TRAINING, PRACTICE INSTRUMENT APCH, AEROBATIC FLT, GLIDER OPS, PARACHUTE OPS, ULTRALIGHT, HANG GLIDING, BALLOON OPS, AGRICULTURE/CROP DUSTING, ANIMAL POPULATION CTL FLT OPS, BANNER TOWING OPS, SIGHTSEEING OPS, MAINTENANCE TEST FLIGHTS, MODEL ACFT OPS, MODEL ROCKETRY, UNMANNED ACFT SYSTEMS (UAS), AND UTILITY AND PIPELINE SURVEY OPS.
To play devil's advocate, a very pedantic argument might be that non-"training" pattern work—a licensed pilot going up for currency, say—would still be authorized. Good luck making that argument, given the spirit of the rest of the prohibitions.
