Ugh, yeah using a modern Mac for this task is certainly not straightforward! First thing's first, what year MacBook Air is it?
You've got all the correct stuff (I assume that #2 and #3 on your list are the Apple-branded accessories? If not, that may be the problem). The 4-pin to 9-pin cable is exactly what you want. The newer 9-pin Firewire 800 port on the Firewire->Thunderbolt adapter is backwards compatible with the 4-pin Firewire 400 port on your camcorder, so you're good there.
Searching on it just now, I didn't find much but did see
this thread over on apple's support forums where a couple other people said they couldn't capture firewire via iMovie on Big Sur. But in their case, the camcorder showed up as a capture source, they just couldn't select it and initiate the capture. Are you saying that your camcorder isn't even showing up at all?
You could also try capturing with QuickTime. Launch Quicktime Player, go to File > New Movie Recording, and in the window that comes up, click the little down arrow to the right of the red button in the middle. Does your camcorder show up there?
If not, try this: Connect everything up, power on the camcorder and get it into the correct mode for capture, and then click the Apple menu, hold down the option key on the keyboard and select "System Information" from the top of the menu. Then go to the Firewire section on the left. If everything is working then you should see your camcorder listed there. If you see "No firewire ports were found" then there is an issue with either the TB3->TB2 adapter or the TB->Firewire adapter. If you see the firewire bus listed there but the camcorder doesn't show up then the issue is either the camcorder (or its firewire port) or the cable you used to adapt 4-pin to 9-pin.
Are you up to date on Big Sur updates? (Apple menu > System Preferences > Software Update) Also check App Store > Updates to make sure you're on the newest version of iMovie.
General troubleshooting stuff--try rebooting the Macbook, and try both thunderbolt ports. Also try it with the mac's power cable unplugged (just a wild guess).