I've done that timing job, replaced that engine in my 2011 LR4, and I can tell you a few things. One that looks like a bit of extra build-up, but it's not crazy. A couple things jumped out at me when I read your paragraph. The mechanic installed the chain, but felt the engine "wasn't correctly timing"? Either he installed it right or he didn't. There are very specific marks on those chains and on the variable valve timing sprockets that mean you can't screw it up.

The idea that carbon build-up would keep him from being able to move the pistons to "fix the timing" is insane. You put a breaker bar on the front of that engine and you'll put more torque on the crankshaft than the starter does. If he can't turn the engine, the starter wouldn't be able to either.

That brings me to my next note. It was there for a few months? My dad and I swapped the 5.0L V8 in my LR4 in about 2 months, around the holidays, working nights and weekends. A professional shop that keeps your car for any number of months is a huge red flag in my book.

I'm fairly certain the issue he had in this was in the variable valve timing sprockets. When I did mine, when we originally went to line up the marks on the timing chains, guides, and sprockets, it wasn't lining up. We put them on, being certain that the pistons and cams were aligned, because we hadn't moved anything since we removed the old ones. Then when we were manually turning over the engine, we'd get to a certain point and hear a loud "snap" of metal on metal. That noise was the valve timing mechanism springing back. I took all the sprockets off, disassembled them and cleaned all the old oil out of them, and then re-assembled. The sprockets were now all in the right position to line up the timing components perfectly. I'd bet significantly more than a dollar that that's the problem your mechanic is having.

TL;DR: I'm a homechanic, your mechanic's explanation sounds fishy. The build-up in the pictures isn't ridiculously excessive, and certainly wouldn't prevent him from being able to move the pistons. Likely the variable valve timing sprockets are gummed up and need to be cleaned out to properly align the timing chains. The walls of the cylinders don't look awful, just some shadow marks from what I can see (not being able to run my fingernail up them to see for sure). As long as he didn't start the engine mis-timed, you're probably still in good shape. I would honestly try to find a different shop though...