Welcome to GMT Nation...
You've probably already thought about this... But jik not... FWIW... Have another look at your Photos for
:
(1) The presence of A Uniform Line of RUST on all Ferrous Metal Surfaces; especially around the entire outer base of the Plated Blower Motor Shell Casing and on the adjacent surfaces of the Body and Door Hinges.
(2) The presence of "The Mysterious Brown Mung"
(Lake Bottom Goo?) you found mashed in between the Plastic Dashboard Fascia segments.
(3) The presence of the Green Cupric (Copper Oxide) Residue inside of ALL of the Plug Wires Contacts inside of the Blower Motor Connector.
(4) The Badly Rusted and Corroded Fastener holding the Rusted Ground Strap Tang to the Mega-Fuse as shown in your Fuse Box Image.
All of these conditions Add Up to imply that the Prior Owner of this SUV was either trying to get a Certification as a "U-Boat Commander"
...or at some time in its life, THIS Vehicle was partially submerged during a FLOOD... and then 'Steeped in it's Own Juices' for quite a while afterwards.
These conditions might explain the difficulties with sorting out THE Most Important Element of
ALL for establishing Good Electrical Circuits:
Obtaining Good Grounds.
THIS Thread from
@Realism will prove very Helpful in Finding, Cleaning and Re-Attaching ALL of the Ground Connections for the GMT360 Vehicles
(with just a Small Dollop of Dielectric Grease added in between the contacts to prevent further Galvanic Corrosion).
While troubleshooting a bad ground I came across some scans of the service manual regarding electrical ground locations for our platform posted on another site. This includes both short and long wheel base, as well as both 4.2L and 5.3L. Credit goes to Wex...
gmtnation.com
One Other Suggestion...
If the Fuse Box Cover is Missing... it will be a Good idea to get a Can of "Canned Air" and after blowing out all of the surface dirt and sand covering the Fuse Box... Pull One Fuse at Time... and do likewise... while looking for any signs of Friction and Vibration Wear on the Plug In Tangs or any Corrosion.
Replace ALL of the Fuses if you find ANY in Very Obvious Poor Condition. The Cost of an entire Automotive Fuse Kit sufficient to fill all of those Fuse Slots is around $15.00 - $30.00 over on Amazon.
Replacement Fuse Box Inner and Outer Covers are abundantly available over on eBay; likewise,
For Not Too Much 'Moolah' :
Get the best deals for trailblazer fuse box cover at eBay.com. We have a great online selection at the lowest prices with Fast & Free shipping on many items!
www.ebay.com
With your Strong Desire to Learn New Things about Automotive Diagnostics, Visit THIS Video on "Multi-Meter Basics" from Paul "Scanner" Danner for some Excellent (FREE) Professional Training that will be very easy to absorb and apply to your need for using a Lighted Probe for Testing Loaded Circuits on your son's SUV and for How To Use a Digital Multi-Meter (DMM) or a Digital Volt Ohm Meter (DVOM) :
THIS Video provides a COMPLETE Free *** Professional Automotive Course on Intermittent Electrical Problems
(which incidentally... covers the REASONS for the Failure you discovered with your HVAC Blower Connector) and so much more:
***
Please Forgive the "Fellahs" designation the 'Old School Instructor' made towards the attending Class Members. Solving Automotive Diagnostic Problems should never be qualified or responses provided based upon a Person's Gender.