There have been photos and links posted about 220V and 3-phase chainsaws, and even some of the industrial, 'package cutters'. But I believe the OP is generally referring to saws running off 120 V household current. This practically limits them to about 15 amps and a 100 foot extension cord, although, I have run them off of a generator doing hurricane cleanup.
Clearly, they are not convenient in the woods, or up in a tree. They are incredibly convenient at a wood pile, where most of the work is done at a sawbuck, or within a few feet of walking. 'On/Off' operation; no rope pulling; no flooding; no smoke; no ethanol issues; no lean seizing; no tune ups; no spark plugs, air filter, fuel filters; etc., etc., etc. Plenty of power. If the saw does not work, you either pulled out the plug or tripped a breaker.
As for the cord, you run it from behind you. Nobody complains about cords using circular saws, reciprocating saws, and lots of other construction equipment. Nobody complains about air hoses running nail guns, etc.
As for the chaps, they are not rated for electric saws, due to the way that they work. We did an informal test at a GTG with a battery powered saw, and the chaps worked by jamming up the nose sprocket on that saw, instead of by jamming the drive sprocket on a gas saw. No guarantees.
Anyone who wants to try a better quality electric saw can rent one of the Makitas at The Home Depot - if you are lucky, they will also let you buy a used one. The Oregon and some Worx models offer moderate priced options. And the $40 saws - well, what do you expect from a $40 saw, gas or electric?
Philbert