The mechanic's comment is that the unburned fuel would eventually wash down to the piston rings and so compromise the engine's compression. Theoretically correct but pragmatically unimportant, IMHO.
These engines don't lack fuel - it's immediately available. What they need is functioning glow plugs and reasonable compression. It's true that extended cranking will warm a cold engine and so help it to start but it's the glow plugs that are the key to starting.
The glow plug relay is mounted on the left fender - one can trace the glowplugs' wires back to the relay. Slide the relay's plastic cover UP - there's an 80 amp fusible link (fuse) there that should be intact or there will never be power to the glow plugs. A replacement link costs about $1 so replace it if it's not intact.
In very cold weather one can and probably should pre-glow the engine two or three times to warm the prechambers prior to cranking.
As to your engine I'd suggest you check the glow plug fuse and replace it if necessary, install a fully charged battery and check the engine's oil level. Then I'd get in the car and the first time only just try to start it like a gas engine - don't wait for glow plugs just turn the key and crank it so the engine turns a bit before it fires. That's to allow its oil pump to distribute some lube oil to the parts before it starts.
Once you begin driving you ought to change its lube oil and filter and watch the clear plastic fuel filter for there might be algae in the fuel that could clog the filter(s).