2011 Base Fit AT:
I’ve had the well-known grind on the first start of a cold day for a few years. I talked to a local shop today, and they said it might be the timing chain. They said they need to hear it to determine whether it’s the chain or the starter. Either way, I won’t get it fixed right away because it’s very expensive.
what are your thoughts?

 

It could be possible, I suppose. This noise started years back. My car may have had 80-100k miles on it (when I replaced the starter because of this fault after moving to Georgia), and it's never sat too long, never been overheated, and never run dry with oil as I've done every oil change myself since procurement at 35k miles (these being some key causes of timing chain, or more specifically, timing chain guide issues)
In Southern CA, a cold enough night was quite rare, but I believe that even way back then the fault existed on the annual 3-5 34°F nights.

I would sooner do the fuel pump than mess with the timing chain. Or do the fuel pump and the starter.
Timing chains are for folks who have experienced the above errors, or who like to do burnouts/launches, or who like to oversize their wheel/tire assemblies to an extent wherein the engine and trans have a shitty time rotating them (see definition of "Jeep")

My Jeep friend is presently experiencing ABS module failure after having already done the chain on a 2017.
I snapped a chain on a Lumina that had both sat and overheated at some point. When I touched the guides on that thing, the plastic-polymer-whatever coating the contact/wear area for the chain to ride against crumbled between my fingertips..

Sitting leads to oil draining fully off much of the chain. then it starts and just rubs hard against those dry guides.
Overheating, particularly as a result of oil loss, just obliterates them.

Mm. This isn't everything that can happen. But. f*ck doing timing chains.