My favorite sentence for this argument is the most simple sentence of all "See Spot run". In your analysis. "Spot" would be the subject of the infinitive "run". In my analysis, "Spot" is the indirect object of "see" and "run" is the direct object.
My objection to this is that it extends the meaning of 'indirect object' somewhat beyond what is normally understood by the label. There is an enormous difference between the roles of
Spot in:
a.
Give Spot a bone. (A direction to someone to give a bone
to Spot.)
b.
Buy Spot a bone. (A direction to someone to but a bone
for Spot.)
c.
See Spot. (A direction to someone to look at Spot)
d. See Spot on the mat. (A direction to someone to look at Spot in a certain place.)
e.
See Spot run. (A direction to look at Spot and what he is doing.)
In #a and #b,
Spot is the indirect object.
In #c,
Spot is the direct object.
In #d, the direct object is not an individual thing; it is
Spot on the mat.
in #e, the direct object is again not an individual thing, and certainly not the verb
run; it is
Spot run, representing what the listener is being directed to see,
Spot and the fact that he is running.
I might be willing to accept your interpretation except that when we change the sentence to "See them run", we have the objective pronoun as a subject problem. I am not willing to go that far.
I assume that 'problem' is a typo. and that you meant 'pronoun'.
Them is not the grammatical subject of the verb
run in exactly the same sense that
they is the subject of the verb
run in
They run down the road. However, as I said with the other sentence, 'them' is certainly not an indirect object. Equally, it is not, on its own, the direct object. That must be, as with the
Spot sentence,
them run. Within that two-word phrase (or, as some modern grammarians would heve it,
clause), it is
them who are doing the running, and the word
them is therefore, within that phrase, the subject of
run.
It is not so odd that a word can function as both subject and object. In
I don't like what is in that message, what is both the direct object of
like and the subject of
is. It just happens that
what does not have separate subject and object forms, so there is no real problem.