Mass Effect does have magic disposal tech, we are never shown it, but the Codex states one thing and they did another in the cutscenes (they never show any radiators or anything get hot on the ships...) in fact the only Sci-Fi game that I've seen to have radiator-"like" parts is AFF:PS... (I believe it would be best if we took this part of the discussion elsewhere for the sake of the thread)
On topic, I think it would be foolish of the GTVA to use fed weaponry, they have been developed for entirely different theaters (GTVA to fight Shivans across multiple systems, UEF for a single one.)
We have infinite beamz as the GTVA, the only good thing out of the UEF arsenal would be the AM-Missile farms, customized for GTVA missile armaments (something like a Supernova MK2 comes to mind.)
All other weapons on the UEF arsenal would be a drag on GTVA tactics as they require ammo, the only thing that would require little adaption are missiles, hence it being the only logical choice, with all that AM it's not unfeasible to thing the GTVA could come up with enhanced designs for Trebs and other missiles.
All of the UEF's (Military) ship designs are useless to the GTVA since they don't fit with their tactics, which would mean time and money spent on making new strategies, refitting UEF ships and other things that would be a waste of resources when you could use them to build already optimized designs.
The part about not quite fitting in with GTVA tactical doctrine is true in a few ways, BUT...those "optimized designs" have been getting completely trashed by the UEF fighter corps. Considering that most Shivan fighters (barring, perhaps, the Mara) are even weaker individually, well...I personally think they'd be stupid not to at least experiment with some UEF craft. It's not as if the GTVA doesn't need logistical support at all. In fact, they're only so dependent on it in the Sol Theatre because they're on the offensive. If they were defending the GTVA systems from a Shivan attack, they'd have the same logistical luxuries that the UEF enjoyed.
It's all about tradeoffs, really. If they save even one destroyer because they spent the time adapting the Uriel to GTVA technology, so that it could rapidly and effectively render Shivan capital ships useless, would it have been worth it? I'm not making those decisions, so I don't know what they'd say. But a destroyer, that's a lot of expensive kit...and more importantly, a lot of not-easily-replaceable personnel.
I've actually been, for my own amusement, going through some of the retail FS2 missions and replacing GTVA ships with UEF ones. I know that it's not that simple, because those missions don't have all the incredibly brilliant scripting work of the BP team, and I don't know crap about FRED, but even without the fancy tricks, it's scary how much more effective the UEF ships are against the Shivans. But, it's against FS2 era ships, which isn't a fair comparison with the post-TEI GTVA. So, I went through some of them again, and used the fancy new stuff. The Atalanta is beautifully capable, and the Nyx makes the weak Shivan fighters piss their pants. In comparison, the Kent is an interception machine. Both the Balor and the Vulcan chew up and spit out bombs. All of the warships have massively improved point defenses, to a point where I had to give the Shivan ships better AI classes and stronger weapons just to make anything a challenge.
I think there's a lot to be said for integrating UEF technology, though. There is nothing like the Uriel for a heavy assault role; nothing in the GTVA arsenal even comes close to being as effective as crippling enemy warships. Plus, despite its size, it's really not all that ungainly - I never had a problem defending myself. 6 Rapiers eat up anything that gets in the way, and Gattlers in the first bank make it good at shooting bombs, too. It does everything, far better than the Herc II, Ares, or Erinyes could ever dream of, and its ability to stay the hell away from enemy AAA beams while it goes about its job gives it excellent survivability. I think that's well worth having to carry around Gattler and Archer ammo - given that the Gattler is a mass driver, and the Archer a gauss cannon, the projectiles are likely very small.
Sending in Narayanas and Uriels against a large strike force of Shivans could eat major threats (large beam cannons, especially) and clear the way beautifully for a hunter-killer group like Serkr Team to jump in at just the right place, with coordinates verified by the artillery units, to wipe the floor with said strike force. The potential for some unique combined-arms tactics is there.