Why doesn’t mes mitzvah push aside shabbos?

Question

If mes mitzvah is doche avoda and avoda is doche shabbos. Shouldn’t mes mitzvah be doche shabbos? Simple logic

If A is greater than B and B is greater than C

Logic dictates that A is greater than C.

Which means mes mitzvah should be doche shabbos

Answer 1

This question is raised in the Talmud:

Yevamot 7a

הדר אמר קבורת מת מצוה תדחה שבת מק"ו ומה עבודה שהיא דוחה שבת קבורת מת מצוה דוחה אותה מולאחותו שבת שנדחה מפני עבודה אינו דין שתהא קבורת מת מצוה דוחה אותה ת"ל לא תבערו

Then he argued: It might be inferred a minori ad majus that the burial of a meth mizwah should supersede the Sabbath, [thus]: If the Temple service which super sedes the Sabbath is superseded by the burial of a meth mizwah, by deduction from Or for his sister, how much more should the Sabbath which is superseded by the Temple service be superseded by the burial of a meth mizwah; hence it was explicitly stated, Ye shall kindle no fire. [etc].

(Soncino translation)

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Edit

Given that the above quote is but one step in a lengthy discussion, and the answer doesn't really make so much sense out of context, perhaps a little more explanation is in order. At this point the Talmud was explaining why the tanna would think that the verse that proscribes lighting fires on Shabbat is necessary to tell us that execution does not override Shabbat. The reasoning is that we could have argued that execution overrides Shabbat via a kal vachomer – if execution overrides temple service and temple service overrides Shabbat then execution should surely override Shabbat. The response to such an argument is that meit mitzvah also overrides temple service yet does not override Shabbat, despite having the same kal vachomer. At this point the above quote comes in to say that maybe in fact meit mitzvah should override Shabbat based on this very kal vachomer (and therefore so would execution). The response to that is that the verse explicitly proscribes lighting a fire.

As explained by R. Yom Tov Asevili in his commentary there, the point is that if the verse explicitly precludes execution overriding Shabbat even though it has the kal vachomer of superseding temple service which itself supersedes Shabbat, then obviously the kal vachomer is not valid. As such the same kal vachomer applied to meit mitzvah would necessarily be invalid as well.

פי' ואשמועינן קרא דלא נעביד ההוא ק"ו ברציחה לדחות שבת ומינה שמעינן נמי לקבורת מת מצוה שלא תדחה שבת בק"ו דהא ההוא ק"ו הא איכא למפרך ביה מרציחה שיש בה אותו ק"ו עצמו ואינו דוחה שבת