I found a couple of shops out of about fifteen from Yelp and Google that would rebuild the present engine for a price north of $3,000.00. Most of the shops don't do that type of heavy engine work so that leaves the dealership or the two independents.

Is it better to rebuild or to purchase a re-manufactured?

again, go with re-manufactured. It's a job best left for the big boys who do thousands of them on an assembly line, with proper testing and quality control. You will know that everything was replaced, not just whatever the re-builder felt like or had on hand. You will also get a good guarantee, and a good shop will guarantee the labour as well. It's good peace of mind.
(by the way, these big suppliers are where the dealerships get their engines. I don't think GM rebuilds engines themselves).

Few of these small time shops know how to do it properly. You never know what you're getting, or if it'll fail a few months down the road because they left a bad part in it.

Thank you I will take at the two companies that re-manufacturer the engines again.

Jasper has a pretty good reputation

Jasper has a good rating on Google and provides a list of installer. There is also Powertrain, Advance Auto Parts, Ebay among others.

I posted several years back on how I resolved the same issues you speak of. My personal mechanic took the time to inspect the entire electrical system bumper to bumper. He found there was a short in the system and fixed it. I have never run into that issue again. It’s been at least 9+ years. I have a 2006 Tahoe.
Have your entire electrical system inspected by a trustworthy mechanic. It solved my same issue.

You should also have seen posts about the "Default" of GM products; which is exactly what you are getting. Even dirt on a sensor, any, can cause the computer system to set off the default you are getting. It will take a higher level diagnostic scanner to start to pin it down.

For Stabilitrack and Traction control to work properly , those systems rely on inputs from wheel speed sensors.

Most vehicles run a static test of the system as soon as the key is turned on and if it senses something is wrong it will let the operator know via the warning lamps.

We don't know if someone worked on those systems recently or if good maintenance was ignored
. A good shop with the right equip and experience should be able to repair this problem without much trouble.

Wheel speed sensors, wiring fault, throttle position sensor, very slightly stretched timing chain affecting the VVT...many things, Any sensor item can cause this GM computer default. I higher end diagnostic system will help pin it down. Most folks don't have one of those. High end technical shop, dealership will. You could get lucky on the first guess try, or start running down the rabbit hole. Unfortunately many shops are pull and replace from best guess.

Best of luck to you

my husband replaced the wheel speed sensor/ ABS sensors behind the break drum and that fixed mine