Carpet Redyeing

No experience but I think it is possible. Nylon and wool are easy to dye but I doubt they can strip old dye first which means they are adding additional dye. There sample photos show going from a light, almost virgin fiber, to dark.

I'm an engineer for a carpet manufacturer, and so I have a dog in this fight. Why don't you just go ahead and buy some new carpet? ;-)
FYI, when carpet is dyed at the mill the dye is "set" using a steamer to hold the material at 212 or thereabouts for 5 minutes or so. Then the carpet is dried using natural gas ovens, which also helps to keep the carpet and dye at a high temperature, further setting the dye. You're not gonna duplicate that process on the living room floor.
Overdyeing is a short term solution, and isn't going to give the same performance. Even if the dye holds up (not likely) most mill-dyed carpets have stain and dirt protecting chemicals applied (e.g. stainmaster, which is a topical anti-stain solution added during dyeing, and Scotchguard, which is an anti-soiling topical solution added later in the process). Both are applied after dyeing. I don't think the spray and pray guy is going to do that...

I thought 3M withdrew Scotchgard from the market, due to fears about perfuorooctanoic acid (PFOA).

They are still selling. To consumers.
See website below....
I had heard a rumor of discontinuation of Scotchguard..... but it was certain spray products were being discontinued and newer ones replacing them.