Safety systems don't want safety systems
Goofus McGee got a job as a mechanic at a big warehouse. $100 million of stuff. In the past Goofus has seen large engines get destroyed because somebody didn't check oil level, so Goofus likes to fit low-oil shutoffs. While making the rounds, Goofus discovers that the fire pump diesel does not have any low-oil-level trip on it. So Goofus fits one on there.
The next day, an employee throws a cigerette into a trash can. No big deal; it's a 1-sprinkler-head fire. Sure enough, the sprinkler opens, the fire pump diesel spins up, and promptly shuts down on a low-oil trip. The 1-sprinkler fire becomes 2-sprinkler then 3 then 10, then the whole warehouse is engaged. Total loss. Goofus says "But hey, I saved a $100,000 diesel engine!"
In fact, you do not put low oil, low water, anything like that on a fire pump. For good reason; you want the engine to destroy itself putting out the fire. Engines are cheap.
So here you are, with AFCIs "protecting" your refrigerator (food safety) and your furnace (freeze/flood safety). And no doubt your radon system and fire alarm. Peachy.
This is a plain case of Goofus McGee legislating safety systems without slowing down to care what it is. We often discuss that refrigerators should not have GFCI, they don't need it and they tend to trip it. AFCI protects wires in walls (and to a far lesser extent line cords, but only where often jostled); I'll argue steel conduit is just as good for a fridge, fire alarm, radon, furnace, etc.
Unfortunately Internet routers tend to be a suitable case for AFCI, since they are so frequently wired up in a rat's nest of power strips and splitters. However again, installed properly, I don't see a problem.
The breakers are tripping because of an Arc Fault
Whatever knocked out your power probably caused the arc fault. Tree falling on wires, or what have you. An arc fault is that kind of thing, and it makes a familiar crackly-snap sound on the wire*, and that's what the AFCI is listening for.
What may be throwing you for a loop is that other FCI device only trips on things that happen on its LOAD side; it cares not about ground faults happening on its LINE side. AFCI is a completely different animal, and it is listening on the LOAD side of the device. However, electrical "sounds" transmit right through (that being hard to avoid) and so the AFCI often hears stuff occurring on its LINE side.
I imagine whatever rended the wires off the poles was a heck of a wallop, so the AFCI breakers would have trouble not hearing that.
You could try putting a whole house surge suppressor, but its job is to suppress surges; the sound of a make-break arcing may have surge components, but it's mostly instantaneous brownouts. Your surge suppressor isn't going to carry your power supply across that.
* Sound is jiggling air, so that sounds wrong. But a "sound on a wire" is electrons jiggling in the wire, in exactly the same motions, frequency and proportion. In fact there's a simple interface to take "jiggling electrons" to "jiggling air": It's called a speaker, and it is that simple. Put 60Hz AC on the wire, get 60Hz audible hum in the air. The air movement and electron movement are in syncopation, and fourier analysis will apply identically to both. So it's not out of line to say you can "hear" spikes.
There's another option for low power appliances
In short: why are you running your generator if you aren't even there?
The simple, cheap solution is a small, always-on DC battery system such as in an old-style off-grid solar home. Most internet router equipment is in fact 12 volts DC, and will cheerfully run off a common small 12V system. Add a battery charger both from the mains-only panel and the generator-only panel (or better: a couple of vertically mounted, south-facing solar panels), and your Internet will be solid as a rock.
It would be nice to graft a refrigerator onto such a system, but the fridge takes 1KWH per day; 1 KWH is a car battery sized thing. So such a setup would be more ambitious, maybe 8 golf cart batteries and 1000W of solar panel for an all-solar setup with no gen.
If you have the gen, that's fine; spin it up, recharge the batteries / warm up home, then shut it down.
What usually puts "getting rid of the generator" out of reach for conventionally built snow-belt homes is the air handler needed for the conventional furnace. They alway use either a huge blower, or a water pump, to circulate heat from the furnace to the rooms. This simply takes too much power; the system would need to be fairly large, e.g. a Tesla PowerWall.
Now in the sunbelt we have a wonderful creature called an Empire furnace. It works on convection and performs normally with the power out, relying on a passive wall thermostat. It can also play with smart 'stats like the Nest (while failing over to a mechanical 'stat when power fails). It mounts in a floor or wall, and can heat both sides of the wall, with 50,000 BTU. The units are in the $800 neighborhood. They are just the ticket as a fallback heater to keep a house from freezing. But they are almost unheard of in the snowbelt because, you know, conventional. But they could be part of a power-loss strategy, especially for an unattended home.