Learn English – Using ‘vain’ in a sentence

To me, the structure of all the sentences look somewhat strange. On the other hand, you are quite right about those rules mentioned. However, we cannot imply them to all the instances.

"Thank god, you reached on time!" or "The train is on time" - I'm talking about particular time; no days/dates required as your rule says!

Now coming to those sentences in concern.

The word that makes all the sentences strange there is need + using a preposition. They all can be written without using prepositions and I think they'll look natural then. The adverb when takes care of that!

Joe and Jill could not reach when we needed them. They reached late.
Good companies reach their customers when they need them.
The ambulance reached when I needed it.

If you want to emphasize the exact time, you may simply use it that way with preposition in.

Joe and Jill could not reach in (exact) time when we needed them. They reached late.
Good companies reach their customers in (exact) time when they need them.
The ambulance reached in (exact) time when I needed it.

Your article is an example of incorrect usage. Arrive cannot be used with to.

Justification: To is a preposition of movement, and arrive does not show movement. You walk to, but do not arrive to.

Prepositions that can be used with arrive are at, in, and on.