3k is the way to go, plus GM designed these engines with a 3,000 mile interval.
"Every 3,000 Miles (5 000 km): Engine Oil and Filter Change (or 3 months, whichever occurs first). Chassis Lubrication (or 3 months, whichever occurs first). Drive Axle Service."
That is from my 2003 S10's owners manual. Synthetic oil was widely available before 2003 so it is not like GM didn't know about synthetic when they built my truck but they still recommended a 3k interval.
Now in reality I don't see an issue doubling it and going 6k intervals on full synthetic oil but I still call BS on that 10k or 20k interval they are pushing now days.
But I don't run full synthetic I was running conventional but have no choice but to use synthetic blend "all mileage" now because conventional has been halted. But even with the synthetic blend the most I would go is 4k or 7 months instead of the 3k or 6 month intervals I always do on my vehicles.
These hills in my state are not easy on vehicles so that is another reason for me.
If you go to the next page in your manual, you'll see the Long trip/Highway interval recommendation is 7,500 miles or 12 months, whichever occurs first. I realize that your conditions call for the more frequent/standard 3,000 mile change, but they did approve an extended interval (conditions apply) with conventional oil in mind.
I also have a 2003 truck, and when I purchased it in 2002, I decided to go 4,000 miles right from the start. I also used Mobil 1 synthetic blend 7,500 mile oil and Mobil 1 filter. I drove mostly steady speeds, but I would also run it to 4,500-5,000 rpm (4.3L 5-speed) a couple times each drive. I followed that schedule until 100,000 miles (4 years), then extended it to every 5,000 miles. By 2009 Mobil 1 discontinued the synthetic blend, so I changed to Mobil 1 15,000 mile oil and extended mile oil filter, and moved to a 10,000 mile interval. Though it costs more than conventional oil, because the intervals are longer the cost evens out. This was near 140,000 miles. I have followed that schedule ever since and am nearing 260,000 miles on my original motor.
Now full disclosure, my motor doesn't smoke/burn oil but I have had an oil leak from the rear of the motor since around 2011. It now leaks about 1-3+ qts between oil changes depending on the outside temperature, so the oil does get somewhat of a "refresh" between changes. I still run my truck hard, but I don't really go past 4,500 rpm anymore.
I think the 3,000 rule is good in general for these and other older vehicles, but I haven't had any issues going the extended intervals in my truck, taking into consideration I have taken care of my truck since new, and what I have already stated.