If Voyager had been Galaxy class, it'd been lame. Intrepid is my favorite ship type, after all
You know, given that we are discussing the performance of fictional ships operating in another part of the galaxy several hundred years into the future, this may be the best reason to pick one or the other. I love discussing the technical mumbo-jumbo, don't get me wrong, but it IS just a TV show.
So what if they had infinite shuttles and torps, plenty of energy and quick repair rates. Had they been constantly out of each of these "commodities", people would have complained how repetitive it would've been for them to be trying to replace them in just about every episode (or miraculously never needing/losing them to begin with) for seven years.
At the risk of dragging this thread in that dreaded direction... If they had been CONSTANTLY out of these things? If they spent time in ALMOST EVERY EPISODE showing repairs, resupply, etc? Hell yes, that'd be repetitive and dumb. But the problem was that they ran to the OTHER extreme, and almost didn't show any repairs at ALL, and just had an endless supply of torps and shuttles without even a throwaway line's worth of explanation as to how they did so. I certainly would never advocate showing even close to every repair job, every resupply stop, etc. But just a few bits here and there explaining how they get all these new resources would have been nice. And when you have an ep in which
Voyager was literally
limping off the screen from damage before the closing credits rolled, I would have liked to have seen the next ep start with SOME acknowledgement of how the ship is back to being pristine.
Well, back to the analysis if it had been easier for a Galaxy class ship... not counting the actual events of the episodes, I think it's a wash. As far as actual firepower goes, I don't think there was much of a difference, actually. Galaxy normally had ~10 phaser arrays originally, while Voyager apparently had ~14 (not all equally powerful, however). The only real advantage would've been the Galaxy's larger torpedo complement. One has to keep in mind that the Galaxy was not a warship either, it was an exploration ship. I'm not sure how the shields between the two compare - Voyager's were probably more advanced technically, but might've had lesser capacity due to smaller ship size and thus power output (speculation obviously).
MA says 13, not 14, for Intrepid. Galaxy is 11, +1 hidden under the saucer. Though... why
wouldn't they all be equally powerful? The Galaxy-class was said to be equipped with 12 "type X phaser arrays." Why would the Intrepid have some arrays that are of lesser strength, especially since it's a newer design?
Regardless, as I said upthread, I don't think number of arrays is a big deal past a certain point. And that "certain point" is making sure all angles are covered, so that the ship has no phaser blind spots. Once that's taken care of, you don't really need too many more arrays, since a single array can fire multiple beams simultaneously.
Although then again Galaxy ships have a lot of non-combat personnel (basically civilians), which in turn is a hinderance in such situations.
Then there's the overall tech level. Voyager was more advanced as far as sensor etc go. Of course this wouldn't necessarily mean much going by the series, since with all the upgrades they did the original sensors had little to do with what they had later on. But initially Voyager would have the advantage. Although here again the higher tech level, while helpful, also makes them a juicy target for raiders that want some new tech. No major advantage here either, then.
With both of these points, I think it would depend on
when the hypothetical Galaxy-class is swept into the DQ. We know there were some upgrades to the Galaxy during DS9, so if one assumed a later iteration of the class (say, the ship got pulled to the DQ instead of
Voyager, but at the same
time that Voyager would have) it could be one of those improved models. However, if you are talking about the E-D itself, during TNG, or another Galaxy built to that spec, it's going to have things a little rougher. Also, as to civilians and families: I always thought having all those civs (and
children) on board was really stupid. One of the biggest series-wide clunkers of TNG. I agree with Ron Moore's assessment, that it could have been (in-universe) a sort of "failed experiment". Starfleet had this idea, and tested it out, but given the realities of how often these ships are thrust into danger (either by circumstance or by Starfleet order), it just didn't work out. My take on things is that after Generations, there are very few civilians on board even Galaxy class ships - maybe a few select officer families (those that have had their request granted, which would be on a much more limited basis), some civilian scientists or technical specialists, and whatever ambassadors happened to be on board. Also the bartender.

But that's it. And NO children.
So if THAT were the case, our hypothetical Galaxy is going to have a much easier go of things; if it's crammed full of civs and kiddies like on TNG, heaven help them.
But as said, in the end I think it's a wash. Both have their minor advantages in different areas, but neither would have any sort of innate decisive advantage.
My opinion isn't too far off. If it's the older, TNG-era Galaxy, I'll give a very slight edge to the Intrepid. If it's the later, improved Galaxy, I'll give it the very same slight edge
over the Intrepid.
PS. I'll say one thing about Star Trek ship designs in general. They rather suck in way, in a technical sense. Looking at how easily systems like weapons etc are disabled, it seems the amount of redudancy systems is very limited. And that's just bad, in all honesty.
Meh, that's the kinda thing I chalk up to "it's a TV show." Sometimes, writers needed to add tension, and thus sometimes would throw in extra things to make the battle seem more interesting, or to make a situation seem more dire. These extra things were not always well thought out. If Trek were "real", these ships wouldn't have so many design flaws. Now, I'm not saying this is an
excuse; I think it's silly how poorly thought out some of this stuff seems to be (the warp core on TNG was one of the worst offenders. The thing would be on the verge of a catastrophic breach if you so much as sneezed near it. And the last-ditch, emergency, "we've exhausted all other options so we're going to use this one" core ejection system was always "offline."

). All I mean is that in-universe, I don't think some of these events would go
quite the way we, the viewers, saw on the show, if it were real.
Three words: Bio. Neural. Circuitry.
Ah yes, the BNC. That is a HUGE point in favor of the Galaxy, if you ask me. This was such a terrible idea. Not only did they not provide any tangible benefit or advantage over "conventional" 24th century computer hardware, but they actually
introduced vulnerabilities! The things could get sick or infected, as we saw, and worst of all, they can get infected completely by accident by Neelix cooking some particularly... uh... we'll go with "flavorful", cheese, which produces this fume that floats through the air vents and slags the gel-packs. CHEESE. The bio-neural circuitry was compomised by CHEESE.
Bio-neural circuitry seems to be (another) failed experiment.