MrAl, hi Again, it seems you are on to something here, may I present you a day's worth of work, I don't know if my testing methods are good enough, but to me, these seem at least sensible I hope it become useful to somebody or at least some odd curiosity to be appreciated
Before I forget, the power supply I used is a 5V capable of spitting out 5A it is a SMPS, gave me some headache with ripple voltage that is why I had to use the caps gotta make or buy a linear adjustable PSU to get a clean signal whenever i get to do this, on this note, for all voltages, please consider +/- 0.2V margin, my multimeters aren't the professional stuff I am a hobbyist.
I'd post a pic of the test rig, but it is so ugly that my schematics and data results will have to do:
Te test Rig for the N-Channel, please pay attention to the voltmeters and ammeter, that is where I collected the results, the thermocouple is a type K cheap stuff but good enough, I used a small clamp to get it close to the FET right on the dain plate, no heatsink was used and on idle the thermocouple shows 27ºC(yes it is in Celsius, if there are any complaints I'll use Kelvin next time Haha)

Figure1: N channel test rig

Figure2: P channel test rig
At the end I included PDF and a spreadsheet file, I used xlsx format, but it was made on libreoffice, I figured somebody may play with it or make charts, not much data, but enough to make sense of things.
Conclusions
Aside from FET #06 that was somewhat not a good performer, but on the datasheet it states that Vgs(th) max is 5V so, that is that... I was actually surprised, I did not tested with different loads, but I did tested with something somewhat worst case scenario for what I intend to do, but I presume that under these conditions even a 500mA load would warrant a heatsink, even small, the lines where the results are with "NULL" are cases where I deemed the test unnecessary due to previous results. Transistors #08, #10-#13 were the best performers, of course #10 is rated as a linear FET, I bought for a project never came through, #11-#13 are P channel, so long as Vgs is kept at -4.9 to -5V it will be fine, but -4.5 things go sour but FET #08(irf540n) goes fine even with Vgs of 4V which was surprising
Bellow is the data of one IRLZ44(Manufactured by Vishay)
Cg = 0.990nF
Rds = 0.11Ohm
Vth = 1.6V
diode voltage=462mV
Vds = 0.175 Id=1.10 T(30s) 33.2ºC @ Vgs 5V
Vds = 0.223 Id=1.09 T(30s) 36.3ºC @ Vgs 3.62V
Here it is interesting to see even with this low Vgs a LL FET still maintains "cool" and low Rds,, too baad I could not find the rest of my IRF540N and I bet there were 3 more IRF95440 somewhere but I can find it later.
with this I can say with confidence that it is possible to use power MOSFETS on 5V Vgs, but only on light loads and maybe medium with a beefy heatsink and like many above said, if it is for repair, it is not worth replacing a signal with power FET. and if it is design, remember to account for the large gate charge and gate capacitor or you might just burn the IO pins on your IC.
To magic:
I see, I wish more manufacturers would put this on the datasheets I remember seeing one with Vgs(th) lines for -55/25/175 ºC but it's been some time, don't even remember what it was. but that is interesting, reminds me of the tubes, but self starting!
Edit: sorry, had to delete and repost because I am still figuring out attachments in here.