By the time I reached home, he had arrived.
Means that you came home, and at that moment, he already had arrived somewhere. The implication is indeed that you found him at home, but he may have arrived somewhere else.
There is nothing to make us believe that he left the place he arrived at again afterwards. It's possible, but there is nothing that tells us it is so.
If you want to say that, you need to do so explicitly:
By the time I reached home, he had already left.
This would mean you (just) missed him.
By the time I went home, he had left for school, but had also arrived moments earlier than I reached.
Here you are trying to put too much into one sentence, and actually it becomes unclear what you mean. It seems these four actions happend, in this order:
- 1) He left for school.
- 2) You went home.
- 3) He arrived (at school?).
- 4) You reached home.
It is not clear what the main point you want to make is. You could simply put this in two sentences:
By the time I went home, he had already left for school.
By the time I reached home, he had just arrived at school.