This is related to what is known as separately derived systems and non separately derived systems, and it concerns avoiding unnecessary impedance on the grounding system that would adversely effect over current protection, i.e., breakers and fuses.
If the neutral is switched along with the phase conductors in the transfer switch, you have a separately derived system and you would NOT separate the grounds and neutrals in the generator only, which contains the main over current device "main breaker". You WOULD separate the grounds and neutrals in the sub-panel. Again, this is because the main over current device, "main breaker" is contained outside in the generator.
So, in your condition the generator is portable and thus does not fall under the permanently installed generator rules. But if it were, you would want to separate the grounds and neutrals at the generator and sub panel and be done with it.
To address your main question whether your current setup is safe. As mentioned above you ideally want the lowest impedance for ground fault protection. But by not separating the grounds and neutrals in the sub panel you are creating MORE impedance.
14 years ago it was surely Code to separate neutrals and grounds on the subpanel. Certainly one would never run separate wires and then bond them; that defeats the point of running them in the first place and sounds incompetent. Aluminum wires installed in the 2000s will be large, at least 4 AWG. They probably oversized the conductors because their installer told them to, the idea being to compensate for voltage drop. There are better ways to do that.
Bailiwick
Part of this is a question of bailiwick. You don't get to fix every wiring defect everywhere, you may have to think globally and act locally. If you want to fix the bonding in the barn, that might be "not your bailiwick" and therefore "not your concern". It is irrelevant as to whether your outbuilding is safe.
Since you seem to have permission to control the outbuilding's subpanel, I would test, by unbonding neutral and ground there (after shutting off power). Try using both as a current return for a hefty load like a heater and watch the voltage across the heater. Both will dip and ground will dip further since it's thinner wire, look for a lot more. Just do it momentarily, this is likely to "light up" all your grounds.
What you want is to have a hot, neutral and ground wire that all work, are separate and and are not bonded anywhere in your outbuilding nor in the cable from the barn. And have grounding rods locally, bonded to your ground wire and not your neutral.
You find wire breaks with specialized equipment, typically by energizing or putting a tone generator on only the test wire, and then wandering about with an antenna listening for the signal (or not). It's possible to find and fix it in a day or two if you call in a pro. So get going. Once properly spliced, it will be legal and "good to go".
Questions
Can you do anything about the missing hot? Sure, you can move the other breaker over one space, so it is also landed on the working pole. If this is a 2-space panel that won't happen, but if it's 4-space, should be sympatico. What do you fill the empty hole with? A cover plate, or a breaker.
Is it safe to continue a suspected-damaged cable in service? Not so much, until you are able to get to it and inspect the damage. Not least, a hot could have insulation damage and be energizing the earth around it.
If the barn has a 30A breaker feeding the long run, your outbuilding's subpanel must be at least 30A in rating. 50A is fine. 200A is fine. You are correct, 30A won't damage a 200A panel :) Your outbuilding needs a main shut-off switch in the subpanel and that's why there's a main breaker at all. It is there as a shut-off switch (or GFCI if equipped).
What's bad about a 50A panel is not many spaces, and you go through those much faster than you'd expect. This situation being a case in point, you suddenly need twice as many because half are dead.
How much power do you want?
It looks like they laid heavy aluminum wire. Modern aluminum wire is fine for large sizes. Aluminum's bad rep actually relates to dissimilar metal corrosion, and the lugs are aluminum, so aluminum wire is the right stuff. Darn shame they ran so much so far, though. The voltage drop calculator is telling me for 30A@1000ft, to use 4/0 aluminum cable (at these sizes using copper isn't even stupid.) That's to carry 7200 watts (!!) I hope they didn't do that.
You can also carry 7200 watts with 15A@480V, or 12A@600V. This will allow 8 AWG or 10 AWG copper to be used, respectively, and stay within 5% drop. You have a bumping transformer at each end to kick it up to the higher transmission voltage. This only requires 2 conductors.
Transformers provide isolation, double-isolation in fact, so your panel is no longer a subpanel, it is a new service which absolutely requires its own ground rods, and neutral bonded to ground in the panel. At this point it really is a main panel.
Assuming they've got #1 aluminum in the ground, they could carry 20KW at 480V (80A@240V) or 30KW at 600V, (125A@240V) at sane voltage drop. That's plenty for a full sized house, and only requires 2 conductors. But you'd really want to fix the cable damage/leakage first!
If it's 4/0 aluminum, that could carry 60KW.
Best Answer
The magic snap is not magic. Contact blades are not flat.
One thing you will notice about US breakers in matching panels is that they "Snap" in with a definite resounding snap. And when you try to remove them, the initial force needed to pull them out the first 0.1mm is considerably greater than the force needed to pull them the rest of the way.
That's because the bus bar in the service panel actually has a little "nub" shaped into it, of a specific shape. That shape engages matching curves on the breaker's spring clip. It works like a detent, creating the snap. However the good contact also creates a lot of mating surface area. As such, it runs at sane temperatures during high current flow, which is why UL approves it.
Every panel manufacturer uses a different shape of bus bar, requiring different shaping on the breaker's spring clip. They are different because of patents. Each manufacturer has patented their shape, because they think it outperforms their competitor's shape (and avoids violating their competitor's patents). Eventually patents expire, such as Square D's QO (queue oh) patent expired, permitting Murray/Siemens to manufacture its QD (queue dee) breakers for QO panels. Still, Siemens is unable to make breakers that fit both Murray and QO panels, because they are still different. They are locked into the old designs because new breakers need to fit old and new panels alike.
So when you use an alien breaker, the clips and nubs don't mate as intended by the two manufacturers, and this is not an accident, it's by design. They are not trying to keep you from using alien breakers, they are (were) trying to keep from being sued for patent infringement. Anyway, you end up with the clips only mating at 2-4 points instead of flat surface contact, and point contact like that is only good for a couple of amps before it starts melting, opening, arcing and burning.