"John Smith" wrote ... the month with the longest day (i.e. June)
correspond with the
warmest month of the year?

Because it takes some time for air masses and bodies of water in the
oceans and seas and, to a lesser extent, large land masses to warm up.
The same is true in winter, when the converse happens and it takes some
time for them to cool down. Therefore, although the strongest and weakest
sunshine in the northern hemisphere are at the solstices (in June and
December, respectively), temperatures lag behind and (as a general rule)
the warmer summer weather is in July and early August, with the colder
winter weather in January & February.

For the same reason, the lead into winter (i.e. autumn) tends to be warmer
than spring, although the elevation of the sun is clearly the same for
corresponding days in each of those seasons.

(Although the elevation of the sun might be the same, the quality of the
light usually looks different during "corresponding days" in autumn and
spring, as well, but that another story).

- Tom.