the chains is an interesting question that I’m trying to figure out.
I had a 2003 Tahoe with Z71 and fully loaded. Really was awesome at getting around on a lake so I never thought about chains.
Now I have a 2014 Silverado that likes to get stuck. I know this is outside your price range but one piece might be relevant. Looked at my manual the other day and it says not to run chains. I don’t if that’s legit or if it’s just a disclaimer.
Where I’m headed is make sure the vehicles you look at can accept chains. Sounds like that might be a deal breaker for you so the lift kit may be the better choice.
the reason chains are not recommended is because you will rip off the anti-lock brake sensor wires in the front along with the caliper hoses,and you can tear up the park brake cable,and or caliper hoses on the rear brakes depending on whether you have drum brakes or disc brakes on the rear.
personally before thinking of running chains,I would run either a softer tire,or lower the pressure in the existing ones to get a better grip on a slick surface,the latter is a pain,but doesnt take long if you plan ahead.
a good quality lift other than running up the keys is the best option as you will maintain ride quality,and you wont wear out the drive train or tires.
the air bags are great as well for the rears,just keep in mind that the air hoses are plastic,and they get brittle when its really cold out and they can snap easily if struck with any form of road debris.
an addition of a posi rear end is a good idea as well,all in with the posi kit,and lift kit will set you back about three grand.
the 5.3 litre is indestructible with good care,it is not unusual to see them last three hundred thousand miles.
the early ones did have oil consumption issues,but that was taken care of and there was a technical service bulletin on that.