When I went to demo my YM186D it seemed slow to start off in any gear. But it engaged solidly within a reasonable time so I bought it.
After replacing the fluid and filter, it engages faster, but not quite as fast as the A/T in a car when you shift from Neutral to Drive. Yes JD303 is what Yanmar recommended 'back in the day'. Later tractor fluids include JD303 among several standards they meet. I use the inexpensive version from Autozone and I see TractorSupply gets theirs from the same source. Some here have suggested that automotive A/T fluid is better for cold climates.
Having some delay in the transmission is an intentional design 'feature'. After you let the clutch out, that starts a little hydraulic pump on the input shaft of the transmission and this fluid pressure then engages the little clutches inside the transmission, which starts the wheels moving. (The pump in the transmission is unrelated to the other hydraulic pump that powers the 3-point).
Similarly, shifting from neutral to L or R sends fluid to the internal clutches, but through a restriction orifice that moderates how fast the little clutches engage.
I suspect this designed-in delay is so the engine can start the inertia of a big tiller instantly as the clutch is is let out (direct mechanical connection), a moment before the fluid pressure builds up to engage the transmission's internal clutches to get the wheels moving. This is a very clever solution to a design challenge. On my older YM240, the torque needed to get a rotary mower spinning is applied simultaneous to starting the tractor in motion. I think this delay in the Powershift allows the equivalent of starting off in a higher (ground speed) gear. Or in practical terms, permits a smaller engine to do the work of a larger one.
Below is
Hoyetractor's picture of the Powershift clutches. (1 and 2 in the diagram below). Half the clutch plates are splined to the central shaft, half to the outer shell. Applying hydraulic pressure to a piston next to the clutch plates pushes the drive and driven plates into engagement.
An automotive A/T is similar but has an additional component for smooth starts: the torque converter or 'slushbox', that has one fan driving another fan which is not quite a solid mechanical link. Yanmar's Powershift has no torque converter so there is a direct mechanical link all the way through the transmission after the clutches are engaged.
Very clever, Yanmar!