I'm going to invoke Occam's Razor on this. While I suppose it's possible that some specific malformed packet(s) could cause your lower cost switches to fall into the failure mode you're describing I'd consider that a very unlikely cause. The switches that you're describing as having problems (small, unmanaged switches) aren't likely to have spanning tree implementations, let alone support for layer 3 switching and dynamic routing protocols. That type of switch should be "blind" to the actual content of the frames its switching, beyond using the source and destination MAC addresses to make switching decisions.
This makes me believe that you had a power issue more widely than you realize.
Going with a power issue assumption, I'd say you're having problems with the low cost switches because they're likely low quality switches. I know this sounds trite, but that's been my experience with networking gear over my entire career (with very few exceptions). You generally get what you pay for (and, though something may be priced incorrectly the market sorts that out pretty quickly).
A higher cost switch is typically going to have a better power supply that is more likely to run within tolerances when exposed to "glitchy" utility power. I suspect that the power supplies in your lower cost switches probably started putting out bad power when the utility power went out of spec. At that point, some part of the "brains" of the switch ended up in a "this should never happen" scenario because one or more of the power rails drifted too far out of tolerance.
An Ethernet switch isn't typically a single ASIC running the whole show but rather are typically groups of systems of ASICs that do different jobs connected to each other. Without knowing about the architecture of the switch in question it's hard to say anything definite. I've had experiences with a model of switch, many years ago, that used a single ASIC to run a group of 4 ports. Certain types of failures would cause groups of 4 ports on the switch to "flake out" while the rest of the switch kept running fine. A partial failure of a switch isn't abnormal in my experience.
In the case of your failure, the parts of the switch that handled keeping the lights on, for example, kept running fine. The physical interface hardware (the PHYs) probably kept running just fine (since you were probably seeing "lights" out on the far ends of the connections). Something else, however, didn't keep working right and you ended up seeing a lack of connectivity. In the cases where I've been "fortunate" enough to catch a switch "in the act" of failing like this I've plugged my laptop into a "problem" port and observed (using Wireshark) a totally "dark" network without any broadcast packets or the other "noise" commonly associated with a typical "working network". Packets transmitted into these ports never showed up elsewhere in the network-- they just fell into a "black hole". I bet you'd see something similar in your situation.