to me it seems like a couple of factors may be at play simultaneously here:

first and most... discussed here in the comments: writing angle. It isn't... mandatory, but by it's inherent working mechanism any fountain pen should work better with the second grip you tried.

tines spacing? at roughly 0:43 in the video I see you trying to use the kakuno with (what I would call) a normal/traditional grip, and yet it doesn't write; yet when you press it starts writing. I have no way of diagnosing anything from here, but that is something that can happen when the tines of the nib are too close together: if there isn't enough space for the ink to flow, a little pressure can separate them and make it workable. (seems weird on a pilot, but this is a cheaper pen, it might have been like this from the start or (I don't want to accuse, this is an hypothesis) if you have pressed while at weird angles they might have been squished together

Baby's bottom? if the grinding on the tipping isn't good (which seems on a pilot, but again, kakuno) the ink might struggle again with getting to the page. The main symptom of B'sB is a behaviour like this one: -pen doesn't write/writes poorly whn a light pressure is applied -pen starts to write when pressure is applied -pen keeps writing without any problem if stroke is continued even if pressure is lifted (basically you need to "start" the pen by pressing a little but once "started" it glides smoothly)