2013 Macbook Air can't get past grey screen
... does anyone know if this machine has diagnostic LEDs like the older iMacs?

There are no diagnostic LEDs on the logic board.

I'm thinking it's definitely a hardware issue at this point but I don't know how to move forward, I figure I can at least recover the SSD.

This is a big jump in determination when two very conspicuous clues were presented:

A prohibitory symbol means that it cannot find a boot volume

Fix a sometimes happening prohibitory sign &#128683;
No entry sign on 2017 Macbook Pro--will replacing the hard disk work?
Can't format my Macbook, HDD and mac Utilities won't load, “blocked” sign after trying to boot usb


A folder with a flashing question mark means the file system on the boot volume was corrupted.

Unable to boot up my Mac. Getting a flashing folder with a question mark

iMac only show flashing folder with question mark after formatting OS X partition drive

Not seeing my HD in the disk utility




Now, I'm not ruling out hardware - it's entirely possible your drive has failed.  But breaking out the screwdriver prior to doing any sort of diagnostics or even examining the symptoms a bit more is putting the cart before the horse.  You can run Apple Diagnostics by holding D  while booting from a powered off state.