I was supposed to do my homework, but I went out clubbing instead. On a literal interpretation, supposed to suggests that other people (or indeed, myself) might have supposed (thought, imagined, assumed) that I would do my homework. I could continue the synonyms (expected, demanded, required,...) but they're getting further and further away from the 'core' meaning of supposed.
I believe that there is some confusion about the core meaning of "be supposed to", FF. If I'm not mistaken, you are confusing two different grammatical and semantical meanings; the lexical verb, "suppose" and the periphrastic/semi-modal, "be supposed to".
So far as I know, the verb to suppose only has the 'duty'-related sense when used as a past participle in this way. Is this true, and if so, why?
"be supposed to" doesn't use the past participle of the verb 'suppose'. It's a standalone structure that's different grammatically and semantically to the verb 'suppose'. I guess that it might have the same relationship to 'suppose' as the word 'use' in "used to + infintive" or the word 'use' in "be used to + ing" has to the verb 'use'. What that etymological connection is, I haven't a clue.
And how come my Mum can't suppose me to do my homework? If she supposed I was doing it, that doesn't imply she instructed me to do it - just that (in the absence of evidence to the contrary) she thought that's what I was doing.
You're right that your Mom can't do that because you are trying to describe a grammatical/semantical structure that doesn't exist. The lexical verb "to suppose" doesn't carry a duty related meaning.
Your Mom can tell you that "you ARE supposed to do/be doing your homework". Note the form of the 'be' verb, "are", that is required with this particular grammatical animal. The use, always, of the 'be' verb, is what makes it different from the use/meaning of the lexical verb 'suppose'.
That meaning is different from your Mom wondering/supposing whether or not you are doing your homework. This meaning is "probably/likely".
The meaning of "BE supposed to" is,
"I am relating to you that you 'must' do something but the compulsive force of 'must' comes not directly from me but from another, usually/often a more powerful force."
OR
"I'm not saying you must do/have to do something, I'm telling you that someone else [or something else] says that you must do have to do something."
(Added later) I think all the above would remain valid and still be exactly the same question if I'd asked about meant instead of supposed.
I don't understand what you are getting at here.