GI's can use damm near any weapon and be good with it based on player skill, as I suspect you know.
Could, don't. Effectiveness matters. ME3 had more crutches than DAMP did as well (shield + health gating, accuracy tied to fps, ops packs, medigel self revives, hydra missles) For example, there were extremely few players that could solo platinum and use none of the consumeables or extras on the loadout (geth scanner being the one of the favorites for solo'ing). Definitely none that did it on all kits.
Good is subjective. Actually effective depended on kit, but no one here should deny some weapons were far better than others, and were used far more than others.
they have wiped the floor with me
However, there is definitely strong and weak loadouts from an objective point of view, nobody can deny this either. Reegar Carbine + Krogan Vanguard. Pretty much any good weapon + Geth Infiltrator. Cerberus Harrier + Turian Ghost are all god tier kits which are very powerful indeed.
Bell Curve, and averages. They wouldn't do as well against themselves, or comparable player with better loadouts, vs very specific cross-kit comparisons (which is why using the same kit for apples to apples comparison is more useful).
Some kits and loadouts were inherently weaker toward the specific player, in addition to players in general as well.
Just because I can, for example, in DAI use Inquisitor weapons with Katari, doesn't mean I will (most of the time). More realistically, I have Sundering available, but I no longer use it because Maul of the Dragon. The same goes for higher tier weapons in ME3MP.
Besides, while I can't recall having used a GI Avenger, I've used the Mantis and Katana on the GI several times on Gold and a few times on Platinum. Also, while I don't think I've ever tried a GI CSMG, the CSMG is one of the best weapons for the GE - and if it works for the GE, it surely works for the GI.
None are 'equally' effective, they're effective at different rates for different people. GI Avenger and CSMG doesn't work nearly as well due to how the TC works (thus the GI works). Mantis and katana are still sniper / shotguns with front loaded damage rather than medium over time or back loaded damage. Just as GI PPR wouldn't work as well as on TGI, Jugg, or even ACA. Though the PPR would technically work, it'd be a horrible idea on the GI. You'd take longer to kill (and yes time is a factor in judging effectiveness), and do less damage.
GE also benefits from tech explosions, where GI does not, when it comes to CSMG and Avenger. While you could use it, it'd still be a bad idea as the weapon rhythm doesn't work well with the GI ablilities.