The only fault I read about these class I amps is you need super clean and plenty of AC from your house since these amps dont store power in the caps or have a massive traditional power transformer to deal with the voltage fluctuations that, they will end up power sagging if you're wall outlet is not up to the task.
I should just let this thread go but I have a hard time letting pass some misinformation. Bear in mind that Class I is the type of output stage, not the power supply. Crown has made Class I amps with "big iron" power supplies, e.g. the K1 and K2, and they have made Class I amps with switch mode power supplies (SMPS), e.g. the CE4000.
Keeping just to the power supply, I believe your point is about SMPS because that's the primary departure from big iron. The first thing to keep in mind is that power supply function is to provide energy to operate the amplifier circuits it supplies. It is the amount of energy it can provide that matters, these days that isn't contingent on big transformers and big capacitors. With SMPS, lots of energy can be supplied from rather small parts because it operates differently than a big iron supply. For example, in my QSC amps, the SMPS recharges the secondary reservoir caps 230,000 times per second whereas in a big iron amp the caps are recharged only 120 times per second. This alone means the caps can be much smaller to provide the same (or even better) filtering/smoothing. In addition, the higher operating frequency means the transformer can be much smaller as well. On top of that, there is also a primary bank of capacitors that charges from the AC mains which feeds the supply /primary side of the SMPS through very low impedance circuits.
While it is true that at some point of low voltage an amp with SMPS may shut down while a big iron amp may keep playing, albeit at significantly reduced output, from a practical standpoint that's really not much of a issue for the average consumer. The exception, perhaps, if you're trying to put on a show somewhere at the end of a woefully inadequate AC mains supply.
Case in point, the CE4000 with its universal SMPS, will maintain full rated output down to ~95V on the mains. A big iron amp will run there too, but at quite a bit less output. At some point lower, which I don't know, the CE4000 will shutdown and the big iron amp may keep going but at even less output. That could be important in some situations, but probably not in your livingroom and probably not anywhere else that has been properly scouted for such an important event where that would be a dealbreaker. To the point, what is the likelyhood you're going to be trying to run 120V amps on 80V (or whatever)?
About sag of the AC supply voltage, that's primarily a function of how much current the amp is trying to draw (and of course the capability of the circuit). Since switching amps are typically more efficient than big iron amps, for any given output power, the load on the AC main should be less, worst case equal. That said, we're talking pro amps here, which usually have more output capability than the typical home amp. So, just by that nature, they must try to draw more from the mains if you ask for more out of the amp. More drawn from the mains equals more voltage sag, regardless of amp style.