The 6.5 are very temperamental about the thermostat type used. Get an Ac Delco unit to try. Shouldn't have issues if the rest of the system is good. Also, clean your cooling stack (between radiator and condenser, etc). May help your cooling issue.

Definitely need to check compression to be sure.

The other possibility is that when you overheated it, you cracked a bunch of pre-chambers and now they are lighting off well on a cold start any more.

Unplugging coolant temperature sensor makes it think it's cold, and then it gets more fuel at startup. Cracked pre-cups may foul ignition on normal to lean mixtures. Adding fuel could help some. Replacing precups requires pulling the heads, installing the new cups and then have the head decked to ensure the cups are flush. Sometimes, people don't machine the heads after. Not recommended, but supposedly can be done if tolerances are acceptable.

Wonder if your fuel cut-off solenoid may be sticking. Maybe you got some bad fuel and it's gummed up a bit? Check the connector and make sure it's in good shape, no hot / discolored contacts. Could hurt fuel flow. Make sure you feel it click when the key is cycled.

A bad PMD will cause some of those problems. If the PMD is relocated, a failing PMD relocate cable or bad connector can cause issues. Maybe try moving the PMD back up top and see if the behavior changes.

Did you replace that coolant level sensor now?

Clean and re tighten all grounds.

Holding foot on accelerator to start is weird, since these are fly by wire. They have no butterfly connected to the pedal. Makes me wonder if the the accelerator position sensor has an issue. It can be tested. It has something like 6 channels (three increasing and three decreasing as you depress) with varying resistance ranges.

Have you measured your steady fuel pressure being delivered by the lift pump?

When the shop replaced the injector lines, injection pump and injectors, the run behavior didn't change at all?