Is there such a thing as a vacuum sump pump? No standing water above check valve?
Sort of, but it probably doesn't work like you think. I'm guessing you want a pump that works like a shop vac. But once you put in a provision to drain off the collected liquid, the vacuum effect no longer works. One could concoct a system that alternately sucks, then drains, but I know of no such commercially available package system. Maybe something like this does exist? I'll bet it would be pretty loud!

There are vacuum primed pumps, they tend to be large, and when the first bit of air hits the impeller as the sump empties, all the remaining liquid in the suction pipe drains back into the sump. Something similar happens with the jet pump you linked to, but since the pipe is small, the volume of liquid drained back is small. But there's always the liquid remaining between the sump floor and the highest portion of the inlet opening, no matter which pump you use. Only manual vacuuming will get at that. In addition, most pump controls shut off the pump before there's any chance of sucking air, leaving even more liquid in the sump.

People typically use self priming pumps to suck out liquid from containers. These sorts of pumps all must retain a liquid reservoir in the casing from which they can re-initiate their prime. There's always some left over liquid sitting around somewhere when ever a liquid pump is involved.

It is possible to concoct a system where after the wastewater is mostly removed, potable water is added and pumped away in an attempt to flush away the remaining wastewater, so that only clean water is left standing. The reality is you are only diluting the remaining wastewater, so some small wastewater portion will always remain.

Jet pumps like the one you linked to are clever devices. You do need good water flow, and they do use a LOT of potable water. Also be aware that the integral check valve of this device will not be considered adequate by almost any building authority. Some sort of vacuum breaker will likely need to be added, further reducing available pressure and volume. These jet pumps are for removing relatively clean groundwater. The bits of food, hair, lint, etc. found in graywater will probably clog the device.