Are you lifting each tub to empty into the dumpster? How big is the trailer you haul the tubs in?
Something I have seen, here in the north, is a system of raising the wheelbarrows, tubs, above the side of spreader or dumpster. One friend has a lean-to on the side of her barn, with floor level 5ft lower than barn floor. They just roll loads to door above spreader, tip manure in. She moves spreader forward to fill the length, as each area under door gets filled. A second friend has a similar set-up, but with a wooden ramp from wide barn door to spreader, to allow wheelbarrows to fill the spreader from front to back before emptying it.
Do you have any low spots to put container in that truck can easily get in and out of? Building a ramp at that low location where you could drive up on ramp with the quad and present trailer, to dump tubs across into container.
Ramp would need to be VERY solid under the weight and vibration of machinery on the ramp. If this is going to be a long term stabling project, you might want to consult a farm builder, who could make a cement pit-style place to put the container, with a cement ramp alongside to dump off of. It would need an incline on both ends to do ALL forward driving. Up the ramp, stop to dump, drive forward and down other end ramp to get off at the other end.
No backing up, EVER.
Unless you have a tractor with a bucket, the dumping trailer contents would need to be dumped beside container, then hand shoveled into the container. Still time consuming, needing to people handle manure twice.
I am just guessing at your set up, my mind picture could be way off reality. No idea of budget to work with for constructing a ramp of heavy wood or rconcrete.
My two friends with the lean-to sheds, have concrete walls alongside the spreaders, very solid. Invariably there will be “incidents” where spreader bumps into the ramp wall. But spreader is covered in all weathers, short distance from stalls to push loads to empty into spreader. For our various weathers, ice, snow, workers are not slipping or getting wet as they work to clean stalls.
I have a 12ft aisle to drive thru our barn with spreader to clean stalls in all weathers, stay dry. Your set up sounds left over from the days when help was cheap, not efficient. 30 horses daily sounds exhausting!! Sometimes I think I am being killed cleaning 8-9 stalls a day, in a very efficient setup! Just whining, not really that hard here.