I see nothing wrong with "They finally agreed to assist the investigation". That suggests that they are reluctant witnesses. Your original "We need more people to assist the investigation" would work in the same way - we need more witnesses. Your "We need more people to assist in the investigation" suggests that you need more investigators.
In English, context is absolutely essential to understanding.
The verb “to investigate” gives rise to three substantives:
1. The gerund – investigating
2. The verbal noun “the investigating [of something]”
3. The deverbaled noun “investigation”.
All three can be used to indicate either the general idea of “seeking a solution to something” or “a specific police operation involving only police officers.”
“In” can mean “within” or “during” or “in the course of”
“With” can mean “attached to [something/an organisation]” or “in conjunction with, but independent of [an action]”
This gives a theoretical 12 or 18 meanings to the sentence, some of which overlap, and this is before we omit “in” /”with”. When this is done “the investigation" can be either an object or a complement.
The differences lies in
1 whether “people” in “We need more people to assist [WITH/IN] the investigation” means “members of the public” or “police officers”.
2. whether “to assist [WITH/IN] the investigation” means “to assist the police/us] [WITH/IN] the investigation” or not
3. what level of assistance is implied by “assist” – an active or a passive involvement.
I just looked it up in my physical Cambridge advanced learner's dictionary. The use is " assist (somebody) in/with something/doing something.
Clearly, in/ with is part of the verb assist as in "look at". At doesn't mean anything here.
Back to assist. As you can see, if the object is a thing, assist requires in/with. So if I go by my dictionary, I must write, a walking stick assists with balance.
Some online dictionaries and Andy write, assist balance. I'm confused.
More investigators do investigating. They are actors. Witnesses answer questions. They are not doing the investigating. Head motion and eye movement working together play an active part in maintaining balance. A walking stick of itself does nothing. It is the hand that uses the stick that plays an active part in maintaining balance. A walking stick can assist balance. Using a walking stick assists in or with balance.
You can say "A walking stick assists with balance". If you do the sentence implies that you mean "Using a walking stick ...". I was answering your question about how "assists" can be used without a preposition.
Some words can be a direct object of assist, but others can’t. These are all valid examples:
Allowing prisoners family access is intended to assist the process of rehabilitation.
This will assist attempts to establish the cause of the problem.
A patient in respiratory distress will also use positioning to assist breathing.
Thank you all, and my special thanks to Andygc.
I always thought assist could take a direct or indirect object without change in meaning. I think I've got it.
1. When Subject assists object, Subject doesn't play the main part.
2. When Subject assists in/with object, Subject does play the main part.
In some contexts both assist with/in object and assist object make good sense, but with different meanings. In others, only one option can be chosen.
3. A walking stick assists with balance. (wrong). It doesn't do the balancing. It helps the hand to do the balancing. So it assists balance. (right)
4. Allowing prisoners family access is intended to assist the process of rehabilitation. = The action of allowing provides some help/ makes it easy for people to rehabilitate.
5. Allowing prisoners family access is intended to assist IN the process of rehabilitation. = Allowing prisoners to see their families will rehabilitate bad people. Some other factors can help.