This is an ancient demo system bug. You must have the same cosmetic loadout as you had it when you recorded the demo.
I did a little test, recording both POV and STV on an offline map. When I was playing, all was well. Unusual and all other cosmetics/skins showed up. Then as I immediately after stopping the recording go and watch it, it's no longer visible in either demo. The spectator window does show the box in which the items are shown (similar to pressing F when looking at a player) so the recording
does know that the items are there, but they just don't show up.
If you record offline they won't show, but if you record on an actual server they will. I have no idea why or if there's a workaround. All TF2RJWeekly ROTW come from tempus servers, so you'll always see the cosmetics in these videos. But you'll notice there are no cosmetics on most JOTW as most people record them offline.
I loaded one of my MG demos done on an online server and the cosmetics show up there. I prefer not to jump online when doing harder maps because for some reason it affects my jumping more than I want to (suddenly interp changes, speedshots have different timings and all sorts of such things) as well as that the map can suddenly move to the next in the middle of me doing a jump.
Is it possible to make my own (simple) server to preserve the cosmetics? I'm not exactly a star in how all of this works...
Update: Hosting a server via
map jump_torii_v3;hostname abc;sv_lan 0;sv_cheats 1;heartbeat did not help either, still no cosmetics.