Air impact wrench, no question. Several reasons.
1. I'm going to have a compressor. What other tool am I going to use to inflate tires and power my air gun? Sure I could use a hand pump and brush but the air gun alone is a good reason to have a compressor. A compressor doesn't have to cost much.
2. Since I'm going to have a compressor an air impact wrench is the lowest cost way to get decent power and in general an air impact will last decades with only a few drops of oil.
Shortly after I got out of school I got my first tools for working on my own cars. It was a socket wrench set, a low buck jack and jack stands and shortly after that my 4 gallon compressor, a 3/8" hose and my $20 impact wrench (later upgraded)
Just using some Kobalt tools as examples:
Under $50 for a Kobalt impact that appears to be based on the old IR 231 design
https://www.lowes.com/pd/Kobalt-0-5-in-400-ft-lbs-Air-Impact-Wrench/1000404901
It only claims 400 ftlb of torque but I suspect that number is less inflated than many and it's actually sufficient for most things people would do with a 1/2" impact. I think an axle nut was the most I've asked of my impact wrench. That was more than my "250 ftlb", $20 wrench could do but the IR 231 had no issues.
We also need a compressor. Well depending on your needs, you may not need a big compressor to drive an impact wrench. Most of the time I've used impacts as burst tools where I need a lot of air only to break the nut free. As such, I suspect even something like this $100 PC pancake compressor would be sufficient for most home impact applications
https://www.lowes.com/pd/PORTER-CABLE-6-Gallon-Portable-Electric-Pancake-Air-Compressor/4764588
These little PCs are light... and noisy. They don't fill that fast as the pump flow rate is low. The similar Rigid compressor I worked on for a friend took about 3 minutes to fill from empty. But with a 6 gallon, 150 psi tank they should have no trouble giving you 10, perhaps 20 seconds of full power into an impact wrench. For a few bucks more the HF 8 gallon hot dog compressors have significantly higher flow rates. They are a bit quieter... "a bit" and a good bit heavier. Still, either of these compressors are well under $150.
Finally, you will need 3/8" hose and a few connectors to get things running. I would assume $25 for those items. So that's about $170 for a basic setup that will run an impact. Some will say you need more and I'm sure some do. In terms of impacts, I've had a compressor similar to this Emglo for 20 years or so. It has never been an issue with my impact wrench.
https://hw.menardc.com/main/items/media/BLACK001/ProductLarge/E810-4V_1.jpg
So if impacts, airing tires and a blow gun are all you need, well a compressor is still possibly cheaper and does offer some features that battery tools can't match. BUT, if an impact is the ONLY reason you would consider a compressor then a battery or AC electric probably makes more sense.