Is my plan for replacing 4x4 deck posts with 6x6s reasonable?
I could ask you to sketch your deck out with dimensions to calc out the beam reaction on the posts, but just shooting from the hip here to save all of us time: there's probably nothing wrong with using 4x4s, especially if they're only 6' tall, spaced that closely, and not also supporting a roof. 

If they're leaning or warping badly, then correct the lean or replace with another 4x4. 

Nobody in my market is using carriage bolts on any new deck, I don't know why that YouTube guy was telling people to. If your beam was a 3 ply instead of a 2 ply, the 1" notch leftover on a 6x6 wouldn't really hold anything. Everybody uses metal post caps from Simpson or USP now. 

This sounds like a lot of work to fix something that isn't broken, all based on a recommendation from a random home inspector. They're usually a jack of all trades, but master of none type of person. I've never had a positive experience with one, they mostly just repeat stuff they have heard and don't really understand.

Home inspectors have to find things to feel like they're doing their job, and if they don't overreact to minor things they might even be liable if something did fail. 

Replace the beam/posts if you want, but if it was me I would add two treated 2x4s to each 4x4 and attached with galvanized 10d nails (8" on center, in 2 staggered rows 1.5" from the edge). One on each side, directly under each deck beam so you have a 2x4 supporting each deck beam and not just the carriage bolts. You'd end up with your 4x4 post looking like it's a 3.5"x6.5" post because of the two 2x4s you added to each one.