Two unrelated circuits drop at the same time
On your original problem
Ed Beal is on the right track.  I think it's a &quot;lost neutral&quot;, but since we now know the circuits are unrelated, I am highly suspicious of a &quot;lost neutral for the whole house.  95% of the time, this is due to a neutral wire failure in the overhead service drop line coming from the poletop to your house, which is not your bailiwick, and which the power company fixes for free in almost all cases.
What happens is that on the service side of the meter (the power company's side), they do not carry safety ground as a wire.  (it's in the air anyway lol).  They use pre-bundled &quot;triplex&quot; wires like this.  The bare wire is the carrier wire which carries the physical weight of the other 2 wires, and is physically anchored at both ends.
the bare wire is used for both the neutral and the carrier wire (we can't do that; the power company can). Years of whipping in the wind tend to break it.  The neutral fails.    When that happens, the two legs are still 240V, but neutral is no longer halfway between them.  So the two 120V legs still add up to 240V, but they are no longer 120/120 - they are 105/135 or really, &quot;all over the map&quot;.  This changes based on the loads of the appliances on each leg in that instant.
This doesn't get as extreme as it could, because neutral has a second path: via the Neutral-Ground Equipotential Bond in your panel, the Grounding Electrode System, a bunch of dirt, to  your neighbor's GES and through their N-G bond.  That is about as effective as you'd expect (better than nothing at all, but not great).  However, this can work  well enough (especially with LED lighting) that people can go weeks or even months without even realizing they have a problem.
Enter the wildcard: the UPS.  The UPS is actually monitoring power quality closer than anything else in your house (including you).  It's going &quot;Hey, this ain't right&quot;.
So my advice is investigate a possible &quot;Lost Neutral&quot;, as it nicely conforms with the stated symptoms.   Get a voltmeter and monitor AC voltage (easy mode: get a Kill-A-Watt energy monitor; its first display is instant voltage.)
Now, try turning on large 120V (plug-in) loads in your house. &quot;Large&quot; means things which make a lot of heat like a hair dryer or toaster.   Watch the voltage. If neutral is in good order, the voltage anywhere else in the house should not deviate by more than 1 volt (maybe 2 at the absolute outside).  Shut off all 120V or 120/240V loads (anything that uses neutral), and call the power company and report an outage.
On that panel
So, driven by this problem, you have a lot of eagerness to fix stuff, but not much knowledge about panels.  You have also been the victim of faulty information sources.  That is fine; we all start there.  Having dealt with plenty of them, here is what I would do if it was my panel and my hard-earned:
Bryant is a perfectly respectable panel type.  In the 80s(?) it was acquired by Cutler-Hammer which has been part of Eaton all along.  In recent years Eaton has stepped out of the shadows and started putting their own brand alongside their various brands, sorta like GM sticking their logos on Chevy cars.  Cutler-Hammer and Eaton hasn't made any radical changes to the BR line since they acquired it, outside of no longer using colored handles (the handle colors match traditional Buss fuse colors).
First, the Square D breakers are gone.  They never belonged in the panel in the first place - there are serious safety issues with crossing breaker brands because the bus stab shapes are different.  UL, Square D and Eaton all say &quot;don't do that&quot;. They get replaced by BR breakers, which are now marketed as Eaton.  Same thing.
As far as being unable to find Eaton BR breakers, that ain't right. BR breakers are the most popular in America.  So you're in an unusual place (like Guam) or just aren't trying very hard lol.
As far as replacing the colored-handle BR breakers, that's really not indicated based on trade experience.  We just don't see reports of old BR breakers failing.  There is a general sense of &quot;breakers wear out&quot;, and I suppose you could swap in new Eaton BR if you really want to... but I'd put it on the safety upgrade D-list, honestly. Other upgrades are more important.
What I would do is turn each breaker off and lift it out of the panel, and look at the bus stabs for any arcing damage, being bent, that kind of thing.  Light oxidation doesn't bother me.  Any stabs that are burned, those get put out-of-service permanently. I wouldn't condemn the whole panel unless problems are severe.
That safety budget
Since you seem to be willing to spend 4 digits on improving panel safety, here's how I'd prioritize that.

Any 3-prong dryer or range circuits get upgraded to 4-prong w/ ground, either via retrofit ground ($10) or using a GFCI breaker ($80).
GFCI protection on all kitchen, bathroom, laundry, garage, basement receptacles, and any water features (pool equipment etc.)  I'd try to do it with GFCI receptacles ($16) with 1 per circuit, because it's cheaper. Keep fridges/freezers off GFCI/AFCI if not Code required.
AFCI protection on any wiring that seems alarmingly old; any bedroom circuits (electric blankets); and any circuits where cords are likely to get mashed against the wall by furniture.  That's what AFCI protection is for.  If the goal is to protect outlets, AFCI receps ($20) are fine since this is retrofit.
An EV charging recep will add value to the house, so I'd throw in a 1&quot; conduit between panel and garage. Enough room for up to 100A charging wires.
Noting the numerous 240V breakers, it's apparent your house has electric toaster heat.  I'd replace both that, and the A/C unit, with a wide-range heat pump that doesn't require separate emergency heat - either straight or split unit.  That would pay for itself in electric bills in a few years, but not right away.
I'd use the heat-pump dividend to replace any remaining Bryant BR's.
I would NOT worry about the &quot;messy panel&quot; or seeming clutteredness of the wires.  Wires bundled tight tend to overheat and melt, so loose is better. Excess wire length is important so you can move breakers around in the future and/or install AFCI or GFCI... so I would not nip them off short like some of the &quot;show panels&quot; you see out there.  Besides, if you ever replace that panel, you're gonna want that extra length.