I don't understand R. Eliezer's view that a zav and zavah who checks her self on their 1st and 7th day are tehorim. Aren't they supposed to count: "Today is day one of cleanliness (no spots or issues), Today is day two..." the same way that we count the sefiras haomer? Isn't this a d'Oraysa? It says "V'safrah la - she should count." Isn't the point of the counting days which are free of zavut-discharge? So if he or she doesn't count each and every one of the 7 days issue free how are they b'chzkat taharah? And this view of R. Eliezer seems to be the halacha. Is it possible that bedikut are not D'Oraysa and that according to the Torah only feeling or sensing issue or discovering it on one's body makes one tamei? (We asked this last point in shiur and the maggid shiur wasn't sure).
B'kavod. Shana Tova,
(1) Tosfos (Kesuvos 72a DH v'Safra; Menachos 65b DH u'Sefartem) does seem to give the impression that a woman ought to count her clean days, as you say, and as the Shelah ha'Kadosh points out (Sha'ar ha'Osios, end of letter Kof). This opinion appears in other Rishonim as well. However, even according to this opinion it would seem clear that the Sefirah makes her Tahor even if she does not keep the Mitzvah of counting orally every clean day. That is why Rav can say that we don't need Sefurim l'Faneinu.
However, the common practice is for the Zavah not to count orally her clean days, and this opinion is recorded in the Rishonim as well (see Shitah Mekubetzes Kesuvos, ibid., Chinuch #330 -- "We have never seen Zavos count their 7 clean days..."; Ran end of Pesachim; Radvaz 4:1102) that the Zavah does not have to count "Today is..." during her clean days.
(2) As for whether a Zavah that does not perfrom Bedikos, but does not feel blood, is Teme'ah mid'Oraisa -- it seems that even after the Hefsek Taharah, since she had a Ma'ayan Pasu'ach she is not considered to have experienced clean days unless she checks at least once, or twice (Rav and Rebbi Chanina) during her clean days. Otherwise, it appears that we would say, "Perhaps she experienced a feeling of blood dripping, but she didn't notice it" (see Nidah 57b and Tosfos DH Katani)
Be well and Kesivah v'Chasimah Tovah!
-Rabbi Mordecai Kornfeld