Is it alright to use logical punctuation?

As I consider the question, it is one of style, that is no question about using an artistic device that one is more or less free to choose among several possibilities so as to decorate the words or whatever, but one of material necessity, and in fact prescribed by logic. However in the English speaking world this misnomer is solidly entrenched in the domain of linguistics and I will use it for convenience.

When you introduce quotes around a block of characters that way you do it in order to suppress in the sentence the primary function that these words have, which is that of introducing referents and coding relations between them and with the referents represented by the other words (not in the block).

• This word is long. (The string of characters is long. The referent of "word" (the particular one talked about, this word) is a word with many characters.)

• This word is "long". (The referent of "word" is "long"; the idea of length in "long" is no longer valid; this is strictly a logical suppression of the normal semantic function of the word in current language.)

In this fundamental way of using quotes, as the quoted block is introduced in a sentence, it must impinge on it as a constituent, in other words, it must make sense. For instance the following is not readily decipherable.

• They talked "the car" because they have time.

The use of quotes involves many principles that are not well known. One important idea is that the practice of using quotes allows to talk about the written word as itself a referent; therefore for transitive verbs that make sense as applied to entities with strings of word (or characters) as referents the quoted blocks are quite natural in object territory.

• He said "Hello". He wrote "Hello". He spelled "Hello". He added "Hello" at the end of all his sentences.

Another likely territory, for instance, is in copulas (subject complement).

• These words look like "a root". This combination of characters looks like a root.

It must be learned at the start that over the ages the human grammarians have introduced numerous exception, or have never considered necessary to apply the principle strictly, and this makes the use of quotes even more difficult.

• […] they call it "the garland"
• He called her Mrs Garland.
• […] he called her "Mrs Piper" (ref)

This brushing up of the basic use of quotes aims at describing their semantic function which is to remove the referents (the semantic function of representing these referents in the sentence) while still keeping to the block a grammatical function. The idea to retain for the present purpose is that the block must be inserted in the sentence as a meaningful constituent.
The commas that you use in your sentence are used to delimit and define constituents, essentially. Suppose that considering the constituent "during the summer" in the sentence

• "They go there during the summer, but never would dare do so in the winter.",

and then decide to place the comma after "the", or even after e in "summer";

• "They go there during the, summer but never would dare do so in the winter."
• "They go there during the summe,r but never would dare do so in the winter."

We obtain pure nonsense. This is why the comma has to be placed after the quote: it is not part of the block, its coding of a pause or whatever in the whole sentence is not removed: it has to be made in the sentence in order to read the sentence properly.

Therefore, the material necessity you perceive is real and there is no escaping it; to do so would be to deny elementary logic, to do yourself and the language a disservice.