R' Samson Raphael Hirsch addresses this question directly in his Siddur commentary, in the "Festival Prayer" section. I believe his explanation is founded on ideas in the Talmud in Berachot and Beitza, so let's start with those.
On Berachot 49a1, the "semi-Tanna" Levi finds it difficult that Yisrael is included in the end of the blessing for the holidays, as that seems to conceptually overload the blessing:
רבי אומר אין חותמין בשתים איתיביה לוי לרבי על הארץ ועל המזון ארץ דמפקא מזון על הארץ ועל הפירות ארץ דמפקא פירות מקדש ישראל והזמנים ישראל דקדשינהו לזמנים מקדש ישראל וראשי חדשים ישראל דקדשינהו לראשי חדשים
Rabbi Yehuda HaNasi says: One may not conclude with two themes. Levi, his student, raised an objection to the opinion of Rabbi Yehuda HaNasi based on the standard conclusions of various blessings. The second blessing of Grace after Meals concludes: For the land and for the food. Rabbi Yehuda HaNasi explained that the conclusion to that blessing is actually a single theme: The land that produces food. Levi raised a similar objection from the blessing that concludes: For the land and for the fruit. Rabbi Yehuda HaNasi explained that there, too, it means: The land that produces fruit. Levi cited two other blessings: Who sanctifies Israel and the seasons. It means: Who sanctifies Israel, who sanctify the seasons. Who sanctifies Israel and the New Moon. It means: Who sanctifies Israel, who sanctify the New Moons.
R' Yehuda HaNasi's resolution for this and other, similarly-formed blessings is that they're really addressing one gift from God, e.g. fruit or sanctified seasons, and mentioning another entity, e.g. the land or sanctified Yisrael, that God uses as the medium to produce the direct object of the blessing. This both justifies the dual form of the blessing-endings and explains, to our purpose, why Yisrael is mentioned in the Festival blessing at all.
This section in Berachot goes on to mention the mixed blessing of "Shabbat, Yisrael, and the seasons," but it's discussed in greater depth in Beitza, so let's go there. Beitza 17a2 explains why the elements in this blessing-ending are ordered the way they are:
תנו רבנן יום טוב שחל להיות בשבת ... רבי אומר אף חותם בה מקדש השבת ישראל והזמנים תני תנא קמיה דרבינא מקדש ישראל והשבת והזמנים אמר ליה אטו שבת ישראל מקדשי ליה והא שבת מקדשא וקיימא אלא אימא מקדש השבת ישראל והזמנים אמר רב יוסף הלכה כרבי וכדתריץ רבינא
The Sages taught the following baraita: In the case of a Festival that occurs on Shabbat, ... Rabbi Yehuda HaNasi says: He even concludes this blessing with mention of both Shabbat and the Festival, saying: Who sanctifies Shabbat, the Jewish people, and the seasons. A tanna taught a baraita before Ravina with a slightly different reading: He concludes the blessing with: Who sanctifies the Jewish people, Shabbat, and the seasons. Ravina said to that tanna: Is that to say that the Jewish people sanctify Shabbat? Isn’t Shabbat already sanctified from the six days of Creation? Every seventh day is automatically Shabbat, without the need for any declaration on the part of the Jewish people. Rather, amend it and say as follows: Who sanctifies Shabbat, the Jewish people, and the seasons, as the Jewish people indeed sanctify the New Moon and the Festival days. Rav Yosef said: The halakha with regard to the conclusion of the blessing is in accordance with the opinion of Rabbi Yehuda HaNasi and as the difficulty was resolved by Ravina.
In establishing the order of the combined blessing, this passage also helps us understand why Yisrael isn't tacked on when the blessing is only about Shabbat: Unlike the Festivals, whose sanctity is dependent on Yisrael's, Shabbat's sanctity is as old as Creation and therefore independent of Yisrael.
R' Hirsch adds some more conceptual meat to the bones provided in the Talmud:
All the festive seasons have had their origin in the history of Yisrael's founding, and their purpose is to help in effecting the fulfillment of the task which has been set the people of Yisrael. It is only through the existence of Yisrael, then, that these festivals came into being, and it is for the sake of Yisrael that they continue to exist.
He goes on to explain how the sanctity of the Festivals is dependent on the Jewish nation establishing the calendar each year by determining the start of each month and when to make leap years.
But the relationship of the Sabbath with Yisrael is quite different. It is Yisrael that had its origins in the Sabbath. The institution of the Sabbath was originally intended to conserve among all of mankind the truths about God and man's relationship to Him. However, mankind had forfeited it. Hence Yisrael was chosen to reestablish this sacred institution, and to restore and spread among men the views of God and man's relationship to Him that are afforded to us by the observance of the Sabbath. Thus Yisrael actually came into being because of the Sabbath, and it is for the sake of the Sabbath that Yisrael has survived.
We thank God for sanctifying Shabbat in and of itself, back in the dawn of time, and we thank God for sanctifying Yisrael, and through Yisrael, the Festivals, whose job is to help promote Yisrael's mission.
Finally, the way R' Hirsch sets up these two relationships for Yisrael as mirror images allows him to resolve a problem brought up in the next part of the passage in Berachot:
מקדש השבת וישראל והזמנים חוץ מזו
Levi cited an additional blessing that concludes with two themes, the blessing recited when Shabbat coincides with a Festival: Who sanctifies Shabbat, Israel, and the seasons. Rabbi Yehuda HaNasi answered: Except for that one.
R' Yehuda HaNasi's answer sounds a bit unsatisfying, but R' Hirsch explains that blessing in light of his conceptual framework: We thank God for sanctifying Shabbat, and through Shabbat, Yisrael, whose job is to promote the lesson of Shabbat, and through Yisrael, the Festivals, whose job is to help promote Yisrael's mission.
1. Thanks to my son, who'd learned it last year in school, for pointing me to this source.
2. Thanks to DoubleAA for pointing out this source in a comment on the question.