I mean hydraulic oil above the inlet. If the machine tilts enough so that a tiny bit of the outlet from the hydraulic tank (where it feeds oil to the hydraulic pumps) is uncovered, the pump(s) will suck air and you stop. I am not exactly sure on your ASV but I believe it's hydrostatic pumps are fed by either a gear (external) or gerotor (internal) pump. Those pumps supply both control oil for the propel pumps and charge oil. In the case you describe control oil pressure is lost allowing the propel pumps to destrokes to neutral and you come to a stop. Towing it to a flatter slope gets oil back where it belongs, the pumps stroke and you are on your way. Machines I've been involved with - we make sure the oil inlet is always covered beyond the steepest slope we expect a person to be operating because oil sloshes around. I feel certain that ASV would be the same. The symptoms sure point to this being the problem though because it worked again once you got it to flatter ground. If it was something like an air leak in the inlet hose(s) the problem should have stayed.
Perkins has been a Caterpillar company since the mid 90's. The engines are the same except for the tags and the color, at least in the size used in the ASV. The engine division applications engineer would visit our facility in Minneapolis, then go on to the ASV plant in Grand Rapids, MN. Then Cat sold its share of ASV to Terex and I haven't kept up on them since. That's why I remember the max slope for the engine with deep oil pan - something customers frequently ask.