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Summer literally doesn't and June is generally hotter than September. Also, who said that in meteorological terms the solstice is midsummer? Try using the solar model for that which is focused specifically on daylight and has summer start in May. That's right there is another model besides astronomical and meteorological.Summer literally starts with the summer solstice.
If as you said the solstice would be the middle of summer, we wouldn't have our hottest days in july and august most of the time.
Even with meteorological terms, the summer solstice still would be nowhere near the middle of summer, it would take place just 3 weeks later.