It seems that, in at least some place and point in time, people really did say "buttload" to mean enough to fill a butt. Specifically, wiktionary lists it (meaning 2) as having been used in Britain, as well as New England and Southern US. However, it's not clear to me which butt unit they're actually referring to, as there are at least 3. And that's before you even consider things like US vs British gallons.
I found this reprint of a source from 1796 which defines a buttload as 6 seams, which is slightly more than 3.5x that of a beer butt: https://www.google.com/books/edition/Reprinted_Glossaries_VIII_Derbyshire_lea/Us44AQAAMAAJ?hl=pt-BR&gbpv=1&dq=buttload&pg=PA70&printsec=frontcover. There are several other sources which define it the same way.
Most other contemporary references I can find to buttload use it as a synonym of "cartload" (a full handcart) in the context of agricultural products such as wheat, not about beer casks. It seems more likely to me that they would stay within the same unit group (bushels, seams) rather than straying into an unrelated one (tuns, butts, hogsheads, etc).
I can't find any usage of the word assload that actually has a specific defined size to it. It's always used as a superlative for having a lot of something. This essay you linked is the only source I've seen so far that equates it as a synonym of buttload (and so by extension, of a butt).
The essay seems to implicitly assume that, because people switched from saying buttload to assload as a superlative, that the previous precise definition of buttload should carry over as well. I'm not convinced by this.
Other sources also list assload as referring to the amount that a donkey can carry, although I have not seen any specific number given for how much weight a donkey can carry.
My current thinking is to add these definitions:
beerbuttload brbeetbutt
buttload cartload
cartload 6 seam