There is nothing ungrammatical about the second sentence. More specifically, there is no participial phrase in the second sentence: there is a subordinate clause followed by a main clause.
In the subordinate clause, “When he arrived”, ‘he’ is very clearly the subject; in the main clause, “his friends met him at the station”, the subject is equally clearly ‘his friends’. This is quite straightforward.
If you use “On his arrival” or “Upon his arrival” instead, there is no subordinate clause at all: there is simply a noun phrase, ‘his arrival’ (in itself just a noun modified by a possessive determiner), which is governed by a preposition, ‘on’. There is no subject or verb in this at all, and no clause.
Using a participial phrase, you do have a clause: ‘arriving’ here still carries enough ‘verbness’ to be considered the verb in a clause (if it were a transitive verb, it would still have been able to accept objects, for example)—but the subject has been left out. When this is the case, the subject will most commonly be automatically filled in by the listener with the subject in the following clause—but, despite Strunk and White’s claims, this is not universal. Some cases end up being ambiguous, and some end up with another constituent taking up the subject slot in the gerund phrase.
“On arriving in Chicago, his friends met him at the station” is ambiguous. The most likely meaning is of course that ‘he’ arrived in Chicago, and his friends met him at the station when he did so. If you go by Strunk and White, though, this meaning is not possible, and the only possible meaning is that the friends met him at the station when they arrived in Chicago; not when he arrived there. In reality, the sentence is ambiguous and would benefit from being worded differently if context did not make it clear which interpretation is true.
Many cases where it is obvious that the subject in the participial phrase is supposed to be something different from the subject in the main clause end up sounding awkward or ill-formed and would benefit from recasting:
Wearing a dark blue satin dress and with her hair loosely draped over her shoulders, he thought her as beautiful as any girl he’d ever seen.
– would be better as:
Wearing a dark blue satin dress and with her hair loosely draped over her shoulders, she was as beautiful to him as any girl he’d ever seen.