Did you check the position of the shift linkage to make sure the neutral safety switch is engaged?

The nss can also be faulty.

It could be just the fuel pump or clogged fuel filter.

What happened to your transmission and what is the current mileage of your vehicle?

Before "firing the parts cannon" and replacing parts, find the issue.

Also by "doesn't start" do you mean that it doesn't crank or that it cranks fast and sounds good but doesn't run.

Get a battery tender on it as you might drain it when we're trying to find your cars actual issue:

1. Plug in an OBDII scanner and check if it has any codes - I wouldn't take those as diagnosis but they're an indication.
2. Check the regular things, Engine oil, Coolant, air filter, Fuel Pressure (does the pump prime in key position II?) does the check engine light turn on in key position II? (an indication that the ECU is online) does your dash display the PRND?
3. Get it on a lift and inspect that there's nothing loose and that all of the connectors are sat in place correctly, that there is ATF, that the transmission isn't seized (in N or D, not in P).
4. Find your car's powers and grounds (for example on AllData) - check all of those and check voltage.
5. Obviously you want to check that relays work, and that both sizes of every fuse (that has a purpose) have power
6. Put your key in the II position and plug in an All System Scanner, it really doesn't have to be OEM but make sure it's an ALL system scanner. Go though your cars systems and check that the ECM, BCM and TCM are all online and that all of the data is correct and changes according to your commands (starter relay, key position, transmission range sensor, ...

After checking, or if you find the cause of the issue, please the results and mention relevant forum members with the results so that we'll get a notification and can try to help more.

If you still can't figure out what's wrong, go to a mechanic. I'm not familiar with big GM products - but they might know something you and I don't. For example, on a Citroen's EGS6 the TCM connectors require special lube or otherwise it causes all kinds of no-start and erratic shifting, or on Volvo's MPS6 the TCM is programed into the BCM and replacement requires programing, I do not know what kind of "surprises" there are in a Suburban, but if you have AllData check what the transmission replacement procedure requires you to do you could've missed something easy.

If all of this is alright, I'd try to find a mechanic who has use a scope to verify can bus communication - someone who's a diagnostics pro and probably not the kind of guy who replaces oil and filters for a living (In recent years those with cars becoming more and more complex - car mechanics who do general maintenance usually can't solve complex issues)

GM cars and trucks are terribly made, On the Cruze and Malibu you're constantly fighting electrical gremlins - and just replacing parts is probably not the solution. You can also watch Scotty's crank-no-start video, and PineHollow's case studies on YouTube are also great, he does plenty of these GM case studies.