Only one person has ever had this section of the 14th amendment used against them since Reconstruction ended: to block Victor Berger from taking his seat in the 1919-1920 House of Representatives; he had previously won a single term in the House in 1910.
This disqualification was done in the House of Representatives itself, with a near unanimous vote holding that he was not eligible for the seat on multiple occasions. Based on events leading to his conviction under the Espionage Act, he was held in violation of the 14th amendment clause in question, and was denied his seat.
Berger (and others) appealed this conviction to SCOTUS, who overturned it in 1921, holding the judge had an impermissible appearance of bias against the defendants. Berger then went on to win three consecutive House elections from 1922-1926. The curious bit here is that no one seemed to have any problem with his taking the seat at this point, and no vote was ever held to remove the disability the 1919 Congress had asserted (as per the 14th amendment), nor any to deny him the seat yet again.
This creates a problematic precedent for anyone looking to deny a seat already won. First is that the enforcement of the 14th amendment provision was done by Congress itself, creating the question of whether that is the only Constitutional way to remove an elected congressperson under this clause. Second is that Congress created an implicit connection between a valid criminal conviction and a 14th amendment disqualification, by simply ignoring his prior disqualification after his conviction's overturning. This is a bit tenuous, but nevertheless a potential obstacle, as the Congressional records for the House proceedings on Berger, a rather fascinating read, are explicit in declaring that they are operating independently of the actual criminal case and judgements.
No court has had opportunity to decide these for certain, but they provide a convenient defense for the accused. As such it is handy if you have alternative and additional paths to pursue your case under, so as to differentiate it. In Marjorie Taylor Greene's case, for example, there are explicit state laws for challenging if a candidate meets all valid legal and constitutional requirements. No such process was used, or perhaps available (I haven't tried to check this), in Berger's case. This creates a substantial legal difference, at least in principle, as it concerns disqualifying a candidate from future seats rather than denying a set to an already-sitting (or at least, already-elected, which SCOTUS has held is much the same thing) congressperson, and is utilizing a non-Congressional legal channel. MTG herself is challenging this law's constitutionality, I presume under the argument that in fact only Congress can effect and judge a candidate's compliance with the 14th amendment. The counter defense, presumably, would be that the case is not attempting to interfere with her currently held seat, which may conceivably be solely under Congress's control, but with potential future seats.
Finally, as a practical matter, the few court cases we've had on denying someone a seat in Congress have all taken about the entire duration of a given Congress (2 years) to resolve. This means the case could become moot before a decision is ever finalized, and even if not the practical result would be a very small loss in the total time the accused Congressperson remains a valid and voting member. So it makes more sense to target all future candidacies instead of the presently held seat, as that will never be formally mooted short of Congressional action pursuant to the 14th amendment.