I've never flown Ag stuff, and I hesitate to engage myself into this fray, but as I read 137.45...
(d)the aircraft at all times remains clear of and gives way to, aircraft conforming to the traffic pattern for the airport. or
in it's entirety
137.45 Nonobservance of airport traffic pattern
Notwithstanding part 91 of this chapter, the pilot in command of an aircraft may deviate from an airport traffic pattern when authorized by the control tower concerned. At an airport without a functioning control tower, the pilot in command may deviate from the traffic pattern if---
(a)Prior coordination is made with the airport management concerned;
(b)deviations are limited to the agricultural aircraft operation;
(c)Except in an emergency, landing and takeoffs are not made on ramps, taxiways, or other areas of the airport not intended for such use;and (d) The aircraft at all times remains clear of, and gives way to, aircraft conforming to the traffic pattern for the airport.
I would imagine the airport involved does not have a tower. I don't have any indication that the operation did or did not engage in agricultural aircraft operation and therefore do not know if (b) applies. It would appear that (a) and (d) would apply.
It seems to me from a basic reading of the account that the AG pilot probably should have complied with the standard traffic pattern recommended in the AIM or facility directory or as established by the airport management. Operating in a reckless manner does not require a regulation to determine if it is or not.
As for the radio, I think a lot of military guys are multi-functional and have mike buttons on the stick. I don't think operating in a traffic pattern requires an extraordinary amount of skill to include a radio in the operation, not that a radio saves you, but it is a tool that the modern age as adapted to. A lot of gliders also don't have radios for various reasons and some older aircraft also don't have radios. Most do.
I have always respected and appreciated the skills of an AG pilot, but really this seems to have been uncalled for as it was described by Mattpilot, regardless of his own problems.