Studying CTY SET kids (SAT math >= 700) suggests that appropriate acceleration doesn't just get kids "to the same place, just sooner", but rather boosts kids onto a higher career trajectory -- more productive, more pre-eminent track -- for their entire careers. (For the kids studied, this was especially true for boys -- for girls in the earlier times, effects seemed to be moderated by the girls' tendency to veer into less mathematically based careers.)
"When Less Is More: Effects of Grade Skipping on Adult STEM Productivity Among Mathematically Precocious Adolescents", Park, Lubinski, & Benbow
https://my.vanderbilt.edu/smpy/files/2013/02/Park-Lubinski-Benbow-2013.pdfThe study was looking at whole grade acceleration, comparing matched kids who did grade-skip vs. those who could have but didn't. I know that's not the situation and question you're asking about -- you're asking not about whole-grade acceleration but rather about a situation where an enriching/accelerating math summer may put your kid further out-of-sync with the school math curriculum. Nevertheless, I bring up this study as it suggests there can be significant lost opportunity resulting from holding mathematically gifted kids (back) "on age track".
