Sounds like you've done the right things. Not familiar with those splitters personally, but looks and sounds like the 2 stage pump is worn. Dirt/debris is the biggest enemy of hydraulic pumps. All pumps wear with usage, and because you don't know the history on this, and based on the brand, it's certainly worn to some degree, possibly very worn.
Engine sounds fine. Can you check the working RPM? More speed = faster cycle times. If you can, shoot for 3600 RPM or even slightly above. You mentioned checking the fluids, but have you changed them? Do some research and maybe try a higher viscosity oil, it will increase pressures, to a degree. Also, can you adjust the pressure relief valve to raise the system pressure?
If it was me, Before I rebuilt or swapped out the pump, as long as you know your filter is correct for the application, I'd swap out the hydraulic fluid, and check that there's no debris or blockage in any of the inlets/hoses, pickups, etc. Hoses can become internally blocked/damaged and sometimes you can't see it from the outside. I've run into hydraulic hoses looking normal on the outside, but being blocked/collapsed on the inside because they break down over time. But yours doesn't sound like a collapsed or faulty hose.
Something else that I can't tell from the video is how sharp the wedge is. I hand file my splitter edges to a uniform working sharpness, and file/grind any high points that will create friction points on my rail, or that will contact the wood or splitter. Verify and clean/lube the rail and slide area. These little things add up and make the splitter work less when operating.
If you drain the fluid, verify that there's no blockages, then I'd swap the fluid for fresh. If that doesn't help, I'd toss in something similar to Lucas hydraulic booster, or some other kind of thickener to see if that helps with the worn pump. There's lots of cheap thickeners if you do some research, and increasing working pressures with heavier fluids can help sometimes. At this point, for what you have into it, what do you have to lose?
If that doesn't do it, pull the pump and see if you can freshen it up, or swap it out.
I'd swap out the fluid first, add some thickener second, file your wedge and lube the sliding areas, check the engine RPM's with a tach. Then go from there.
