The difference between "…to {get, arrive, reach, come} here on time."

Question

This is a B2 question taken from an old English test paper. The student has an original sentence and is given a new word which they must use to create a new sentence–using between two and five words–without changing the meaning of the original.

There's no chance of Jenny getting here on time
POSSIBLE
It won't be _______________ here on time.

A private student of mine who's Italian, wrote

It won't be POSSIBLE for Jenny to come here on time.

My book says the solution above earns the candidate one point, in order to earn two full points the correct answer must be:

It won't be | possible for | Jenny to GET/ARRIVE/REACH here on time.

I'm not disputing that the book's solutions are correct, I disagree that "come" isn't included among the options.

With the preposition "to", come and get both work

1. I COME to work on time. (YES)
   I GET to work on time. (YES)
   I ARRIVE / REACH to work on time. (NO)

The verb arrive can be used in conjunction with the preposition "at":

2. I ARRIVED at the airport on time (Yes)
   I CAME / GOT / REACHED at the airport on time (NO)

When there is no preposition, all of the verbs below are acceptable.

3. I usually COME home on time. (YES)
   I usually GET/ARRIVE/REACH home on time. (YES)

Why does Cambridge, the ESOL examining centre, consider "come here on time" incorrect in this instance? Does the meaning change?

Answer 1

I see coming to a place as referring to the action of travelling there, while what happens 'on time' (or not) is the arrival there.

I come to work on time isn't ungrammatical, but I find I get to work on time or I arrive at work on time much more natural utterances.

Answer 2

COME and GO work differently semantically than GET/ARRIVE/REACH, and I am not taking the prepositions into account here.

Those two terms involve issues of directionality and speaker location that get, arrive and reach do not.

I will only go into one example, as it can be complicated:

My boss is at work or in her mind is thinking as if she were at that location.
We're talking on the phone.
My boss says: Can you please come to work on time tomorrow?
Answer: Yes, I can come to work on time tomorrow.

The Cambridge ESOL probably was taking the idea of location and speakers into consideration as that is the only feature that separates come usage from get/arrive/reach usage.