Have to admit I've no idea, not being able to look at your machine, but since you report a belt between the hydro and engine and mine doesn't have one, I'm guessing that there is a design difference between your 721DT and my 623T. What year is your mower? I purchased mine new this year, but it was built in 2017.
What I see (once a protective cowling is removed) is that the 623T Kohler Command Pro engine has both a front and rear shaft. The front shaft and the tandem hydraulic pump shaft are assembled parallel and directly in line with each other (i.e. end to end). Not sure how big the gap between the shafts is because there is a floating but rigid cylindrical coupler which pretty clearly serves as a limited universal to allow for any small shaft misalignment. By floating I mean that the coupling cylinder feels relatively loose, and can easily slide back and forth about 3/16". I haven't looked closer, but undoubtedly there is an interior key or splines on both shafts which "lock" rotation of both components. In any event, I have no "small belt behind the hydro" like you report, so your machine is certainly different in that regard. For completeness the rear engine shaft on my 623T powers the PTO, but that's likely incidental to this discussion.
I would suggest checking if your engine and pump shafts are offset and if the "small belt" you report connects pulleys that are in line between those two shafts. If so, that would just be a different approach to connecting power between the engine and your hydro pump, in which case that belt becomes the prime suspect for your problem. If it got thrown, broke, or started slipping, it would certainly cause both wheels to become inoperative at the same time.
cpurvis suggested the complete pump may have died. I'm not familiar enough with the tandem pump design to know if that is likely, but if the belt hypothesis above does not pan out, then from your reported problem I would still focus on components common to both pumps. That's because the separate left / right pump sections, while co-housed in the tandem pump chassis, would probably not both die at the same time (that is, that possibility would be the last thing to check on my list). The common components which would more readily explain simultaneous failure are limited to the shared drive shaft, a common fluid reservoir, and a common filter, so be sure to check that the shaft did not break (or freeze up), that the fluid level is not low, and if both of those are OK, that the filter is not plugged. Unfortunately, I don't think you can check the filter without draining the hydraulic fluid, which is expensive. If someone is aware of additional common components to check, please chime in.
The note by 173abn that the pump on the 721DT might be powered via the PTO puzzled me. There could certainly be unusual designs out there, but isn't the 721DT PTO switchable on/off like most mowers? If so, and the pump was also powered by the PTO shaft, then you wouldn't even be able to drive unless the PTO was enabled, which to me seems counterintuitive from a safety standpoint. Or am I missing something?
Good luck with your troubleshooting!