I'd say that the most important thing to get right when looking at the area of transitivity / intransitivity is to realise that many verbs have both transitive and intransitive usages.
Some verbs are strictly intransitive (eg arise; arrive; chat).
Some verbs are strictly transitive (devour; interject).
Probably, most take part in both sorts of constructions, sometimes with different senses
(the warm wind melted the ice; the ice melted)
and sometimes with the same sense
(I haven't eaten yet; I haven't eaten my tea yet).
This latter category includes some constructions where a normally intransitive verb is used with a direct object that is part of a very restricted set:
John danced well / John danced the most difficult dance on the
programme / John danced a tango.
The second point here is that the construction I've got an N to V is not forcing 'objectlessness' on the verb V. 'I've got a man to see' implies 'There is a man I must (or may) see.' / 'I must / may see a certain man.' 'Man' has the thematic role 'theme' throughout (the one who is met and 'seen'). The to-infinitive is a 'completer' (it may be easily dropped, as in 'I've got a book [to read]' (again two meanings here: necessity – when 'to read' meaning 'that I must read' is necessary – or plain statement of fact) or not, as in 'I've got a retired chef to interview', 'He's a hard man to track down'(!) and the idiom 'I've got a bone to pick with you').