This webpage, under Specific Time, says that 6月 in 6月は涼しい is the subject of the sentence and thus does not take に, but that's like saying 日曜日は忙しい and ignoring the implied 私が. In the former case, that would be 天気.
に can specify time. This is a unique function of に. However, don’t confuse it with the topic maker; は. For example, with this sentence; ６月は涼しい (June is cool) , ６月 is the subject and hence you cannot use に.
I don’t think it’s correct to say ６月 in that sentence doesn’t take に because it is the subject.
As you suggest, ６月 is clearly not the subject In the following modified sentence but it still doesn’t go well with に. (That’s if a sentence can contain only one subject.)
６月は気温が涼しい。
I don’t think it is even necessary to go into that discussion of whether ６月 is the subject or not. It doesn’t take に because the sentence has no verb. When a time expression is marked with に, it indicates that some event happens at a specific time. And the idea of some event happening is usually expressed with a verb. A nominal predicate makes the sentence sound incomplete as we discussed here. An adjectival predicate makes it even worse, if not outright incorrect.
x ６月に涼しい。
On the other hand, the following sentence is perfectly fine.
６月に（は）涼しくなる。
Here, the verb なる indicates that something (a change) happens at the time marked with に.
When に is used with an adjectival predicate, it usually carries the same meaning as に対して.
環境にやさしい。
It obviously doesn’t mark a time expression like ６月 in this case.
This doesn't directly answer your question but I think whether ６月 is the subject or not is moot here.
日曜日は忙しいdoesn't necessarily imply 私が. It can simply be "Sundays are busy (days)" without involving any other subject.