Vastly easier to synchronize 2+ generators
With DC generators (or AC alternators immediately rectified to DC)... each generator simply has its own voltage regulator set to output the agreed DC voltage. The regulator, which is simple, well-established tech, simply increases/decreases the generator excitation field to get the correct voltage to come out. It will do this at any engine or generator speed. Your car already does this with its alternator.
The only issue is on multi-engine vehicles. Starting an engine draws a tremendous amount of current. The generator on the already running engine might overload itself trying to help start the other engine. However this is easily solved by adding a current-limiting circuit to the regulator, and that can be done entirely with passive components.
Whereas synchronizing AC is super hard
When you have AC distribution, and you want to feed that distribution from 2 generators, it gets byzantine fast.
First, the generators must be synchronized to each other before the second one connects -- both in frequency and phasing. If not, it's a really bad idea to bring them online.
If the frequency is not matched, extreme current will flow, as each generator tries to "force" the others to run at its own speed. Imagine 2 engines connected with a clutch: they're at different speeds and then you abruptly "pop the clutch". The engines will be forced into sync, and the forces on the clutch will be tremendous. Well, the same thing happens, but with electric force which will be tremendous. Likely, this will simply trip their circuit breakers.
If frequency is matched but phase is not, the generators will still fight each other, as each tries to maul the other into sync. It's like if the engines were running at the same speed, but the clutch had gears and the teeth weren't aligned. There will be a big shock, and again that will occur with electric force onto the AC buses.
Synchronizing is straightforward as you saw in the linked video. But since you don't have a flight engineer to watch the 3 lights and turn on the generator at the right instant, this all needs to be automated. Not so simple - lots of complex things to go wrong.
Once synchronization has been achieved, there is a reactive force flowing between all generators to keep each generator in sync and phase-lock. If a generator starts to overspeed, its share of total load will increase, and this load will make its engine bog down. If a generator underspeeds, load will come off it entirely and it will speed up. At least that's how it works with engines *whose primary load is the generator. The tail wags the dog: the grid wags the engine.
... even harder when engines must be free to run at variable speeds
You caught the part above where the 2+ generators will "fight" to force each other into sync. If the generators are hard-coupled, this means they are trying to force the entire engines into sync also, like an electric version of the cross-shaft on a V-22 Osprey.
That "tail wagging the dog" is fine when the engine's main output is the generator. But the plan falls apart if the generator is an auxiliary load.
So that means thrust-generating aviation engines can't be hard-coupled to their generators. There must be a variable-speed mechanical coupling that frees the generator to run at its ideal/target speed. This is more complexity still, and more stuff to break.
This concept of a 'mechanical coupling' also exists in the DC generator, but this happens electrically -- the voltage regulator adjusts field excitation to suit engine speed and electrical loads. That's what voltage regulators already do. Easy as pie!