All good points. I'd set it up very near the airport but not on it and then access it using the same web browser that I'd use to go to ADSB Exchange.
The problem is, you need to have a good height for the antenna - "height is might" in radio, particularly above VHF bands. I actually can see this with my own ADS-B receiver - I'm in a valley and precisely can see that effect when plotting received packets.
It's not something halfway critical to the operation–why would the FAA allow that? ADS-B Exchange is not run by volunteers–it's run by employees of JETNET LLC, an aviation intelligence company. Doing it yourself almost certainly gives you less information–you're not part of a global network of receivers. It almost certainly gives you less reliability–receivers in the big networks typically have a fair amount of overlap which gives redundancy your single receiver doesn't have.
It's also not FAA approved!