"At midsummer" would refer to the time of year. The capitalisation of the quoted answer, shows that this is referring to a specific day to celebrate the summer solstice. "On" would be appropriate for a particular day, just as we would say "at Christmas(time)" but "on Christmas Day".
Personally I have never heard a specific day called simply "Midsummer" in English. "Midsummer Day" (or "Midsummer's Day") is the 24th of June, one of the four "quarter-days".
So, the translation capitalizes "midsommar" into Midsummer, which clearly refers to a specific day in June. However, when reading it in Swedish, it's (of course) not capitalized. So, could I then understand "på" to mean 'at' midsommar, more broadly as a period of time in the summer than just the one special day?