Yes, the Sadducees did believe they were Jewish.
However, the confounding factor in the quotes you provide is probably this: That in many gemaras, because of medieval censors, "Christian" or rather "Min" (Christian sectarian) was replaced with "Sadducee".
See for example the London manuscript of Yoma 56b that you cited above. I have drawn a red arrow to the word מינא:
So too in the Munich manuscript on the same daf in Yoma (I've included the right margin, so use that to locate this snip in this larger image):
The idea behind replacement theology is that God has removed himself from the Jewish people for being so awful, and has therefore replaced them with a new nation, namely the Christians, in terms of the Abrahamic Covenant.
As such, verses speaking about how awful the Israelites were such that God has abandoned them are precisely the sort of verses they would cite to prove the replacement theology. It would not be self-criticism to cite these verses. These verses would presumably not be targeting the new nation within the new covenant.
Thus, in Eruvin 101a, Michah 7:4 reads טוֹבָם כְּחֵדֶק, יָשָׁר מִמְּסוּכָה; יוֹם מְצַפֶּיךָ פְּקֻדָּתְךָ בָאָה, עַתָּה תִהְיֶה מְבוּכָתָם, "The best of them is as a brier; the most upright is worse than a thorn hedge; the day of thy watchmen, even thy visitation, is come; now shall be their perplexity." That only the prophet, who is כְּאָסְפֵּי קַיִץ, "the last of the summer fruits", is righteous. But among the people, even the "best" is really a briar. Rabbi Yehoshua reverses the ascribed meaning, such that the righteous do exist and their merits protect us. Or alternatively (and this is polemics, mind you) they send the other nations of the world to Gehinnom.
And in Yoma 56b, it is all about sin and rejection by God. The verse in Eicha 1:9 reads, טֻמְאָתָהּ בְּשׁוּלֶיהָ, לֹא זָכְרָה אַחֲרִיתָהּ, וַתֵּרֶד פְּלָאִים, אֵין מְנַחֵם לָהּ; רְאֵה יה אֶת-עָנְיִי, כִּי הִגְדִּיל אוֹיֵב, "Her filthiness was in her skirts, she was not mindful of her end; therefore is she come down wonderfully, she hath no comforter. 'Behold, O LORD, my affliction, for the enemy hath magnified himself." This is a reference to the sending of the Judeans into exile. The Christian says השתא ברי טמאים אתון, you are surely impure. As Rashi explains, the intent is ודאי טמאים אתון ואין שכינה ביניכם שורה בטומאה, you are certainly impure and the Shechina does not dwell amongst you to dwell in impurity. In other words, this is a rejection of God among the Jews. The response of Rabbi Chanina is that God dwells amongst them even though they are impure. Thus, אמר ליה תא חזי מה כתיב בהו השוכן אתם בתוך טומאתם אפילו בזמן שהן טמאין שכינה שרויה ביניהן. He said to him, go see what is written about Him (in Vayikra 16:16), השוכן אתם בתוך טומאתם, Who Dwells with them amidst their impurity. Thus, the figurative impurity does not represent a total rejection of Israel.