Other Semitic languages have the ‘neighbor’ word for ‘foreigner, stranger’, e.g. Hebrew ger Syriac/Aramaic giyyura, Classical Ethiopic gəyyur (from GWR ‘to sojourn, dwell (together), live’), but GRY originally meant something like ‘to incite, stir up, attack’, compare in particular in Akkadian gerû(m)/garû(m) ‘to be hostile to, attack’. The meaning ‘to run’ is found only in Arabic, but could be developed from ‘to attack, stir up, agitate’ > ‘to cause to/make run’ > ‘to run’.
However, ‘be hostile to’ and ‘stranger’ are so similar (compare guest, host and hostile, all going back to the same word meaning ‘stranger’, https://www.etymonline.com/word/*ghos-ti-) that I think that there was an ancient connection between these words.
Erin, I don't yet know enough about Arabic, but if Arabic does distinguish between gerunds and infinitives, it will never count that correct. If English greetings and infinitives are both always translated into the same form in Arabic, then your option should be added as an alternate. I hope am Arabic speaker might clarify.