There is a common source put forth for both arrival and intercession, and it is the concept of contact (=נגיעה). The word appears in parallel to a number of other terms of entreaty in Yirmiyahu 7:16. There, Shada"l addresses (pp. 28-29) the question of what a term of touching is doing in the context of supplication.
וְאַתָּ֞ה, אַל תִּתְפַּלֵּ֣ל בְּעַד הָעָ֣ם הַזֶּ֗ה, וְאַל תִּשָּׂ֧א בַעֲדָ֛ם רִנָּ֥ה וּתְפִלָּ֖ה. וְאַל תִּפְגַּע בִּ֑י כִּי אֵינֶ֥נִּי שֹׁמֵ֖עַ אֹתָֽךְ
Shada"l explains that the relevant phrase means "don't beg me", invoking a term of physical contact like the kind one might make while trying to persuade the object of their plea to fulfil a request. (I assume he means that contact to be figurative, but maybe not.)
So the sense of arrival would derive directly from contact as one's making contact bodily with a place. And the sense of intercession would be a less direct kind of contact that one makes in the course of approaching the one to whom they are interceding.
At the end of the comment Shada"l refers the reader to Bikurei Ha'itim 5589 (p. 93 of this PDF; beware that the pages are in order that an English book would be so you have to skip up and down), which says that פ.ג.ע is in turn derived from נ.ג.ע. These both relate to two things coming in contact whereas their counterparts, פ.ג.ש and נ.ג.ש, have to do with two things approaching but not touching. In this way, the other senses of פ.ג.ע are all explained as forms of contact, for better or worse:
- meeting, as the angry mob did to Moshe and Aharon on their way out of an audience with Par'o because they constricted their ability to continue on their way.
- meeting, as the manner of encounter between a manslaughterer and his victim's relative, in which vengeful physical contact is made.
- attack, by striking.
- bordering, in which one territory touches another.
- interceding, as in Avraham soliciting help from the Chitim for his appeal to Efron. The more direct/proximate verb פ.ג.ע is appropriate for this type of appeal, where one knows that the recipient will be unwilling to part with the object of the request and therefore, presumably, needs a no-holds-barred pleading.