Gill, Spurgeon and others lean toward cessationism as do some Christians and also as do some within the Messianic Judaism movement. Thus the reason why Gill may not credit 'his innermost being' as being the indwelling empowerment of the Holy Spirit (e.g. 1 Corinthians 12:7-11) in a faithful Follower of Israel's Messiah. However, accordingly cessationism theology didn't take place until the passing of the Apostles years after Yeshua's own words (John 7:38). So, Gill's logic doesn't hold water with respect to Yeshua's own words in John 7:37-39; especially for those that hold to the theology of supernatural Spiritual continualism being just as active today as at the time of the Apostles.
Agree that John 7:38 is most likely a reference to Zechariah 14:8. However, that is not to say that [Spirituaj] living waters can't flow(metaphor/allegory) from a born again Believer having the indwelling presence of His Spirit (John 7:39). Thus, the use of both His as well as his is certainly plausible. Certainly understandable to the early Messianic Followers of Israel's Messiah being the physical manifestation of Incarnate God. You may prefer that translators capitalized His instead of his when reading the following translation ... ‘From his innermost being will flow rivers of living water.’ The NKJV uses capitalization as follows telling the reader who is Yeshua and who is a Follower of Yeshua ...
John 7:37-39 (NKJV)
37 On the last day, that great
day of the feast, Jesus stood and cried out, saying, “If anyone thirsts, let him come to Me and drink.
38 He who believes in Me, as the Scripture has said, out of his heart will flow rivers of living water.”
39 But this He spoke concerning the Spirit, whom those believing in Him would receive; for the Holy Spirit was not yet given, because Jesus was not yet glorified.
John 7:37 (CJB)
Now on the last day of the festival, Hoshana Rabbah, Yeshua stood and cried out, “If anyone is thirsty, let him keep coming to me and drinking! (me should be capitalized)
Gill's (and your) preference may be for the translators of Vis' red quote to have capitalized His in John 7:38 implying Yeshua's (His) 'innermost being' instead of a Faithful Follower's (his) 'innermost being'. And yet both are applicable today as they were during the time of the first Apostles.