I tried these questions with cram. The する ones seem to accept いたします as correct with no further comment. The いたす ones don’t accept します but the answer isn’t marked incorrect; I was only asked to make it even more polite (which is an obvious hint that I should use いたします).
In other words, simply choosing the wrong option will mostly not result in being marked wrong. But these things can:
- Choosing the wrong honorific prefix (お vs ご)
- Applying a prefix when it’s not used with the word in question (e.g. with words that are already humble enough like 拝見 etc. or 失礼)
- Applying a prefix when it can’t be used because the action doesn’t involve the person being shown humility (e.g. ご勉強 is wrong because studying doesn’t involve anyone else, while 勉強いたします is possible with いたす as a humble form of する)
- Not applying a prefix with いたす when it’s not simply replacing the verb する.
For some reason, the いたす questions only seem to produce an orange warning for wrong prefix + いたす. In a way it makes sense because they’re saying “at least you remembered いたす,” but I think it’s also logical that believing an honorific prefix to be appropriate would lead to attempting to answer with する, so it’s kind of the same mistake…
I agree that sentences like these are difficult to interpret without context. “I’ll close the door” is such a basic thing that I wouldn’t even consider using a humble form or even a polite one by default if bunpro wasn’t explicitly asking for one - there isn’t even enough context for that. In these sentences, お+stem+いたす is more or less really just a more polite way to say the same thing as お+stem+する. (Again though, this sentence does actually seem to accept both options…)
I don’t really see a problem with being prompted for a more humble answer by the いたす sentences. It’s not the only grammar point where bunpro does something like this because many things can be said in multiple, equally correct ways. But it would be nice if more sentences where like the one about sending the merchandise, since that one is clearly in a business context where we’re talking to the customer, which is a pretty good hint that we should use いたす if we can.
(Note though that there is an out-of-band hint: The いたす questions want a “[humble]” statement, while the お+stem+する ones want to do things “humbly”.)