Hello ‘jelly-bomb’ and welcome to the forum. Interesting name btw.
In a word, yes you should be able to put brake fluid straight into the reservoir after removing the cap. I don’t have a gauze or anything at the neck of the tube, but after a quick look at the electrioic parts catalogue (EPC, see: https://toyota.epc-data.com/regius_ace/lh119v/114894/chassis/4702/47299/ for my van) I see there is one (part 47299-26020) so I probably should get one of these. All I can think of is maybe the one in your van is a bit gummed up. These can sometimes restrict refilling screenwash when the fluid there goes a bit weird but I’ve not seen this for brake fluid. All I can suggest is to have a poke about and see if this might be the cause?
By the way, you might like to take a look at the longish post at the top of my ‘show us yours’ page where I’ve put a fairly detailed post on various resources. Happy if it’s any use to you.
HTH!!