The Mishna 68a and Braisa 68b refer to the hefsek tahara as being "hifrisha b'tahara", and is certainly essential to being able to go to the mikva. However, the mishna and gemara only clearly demand this in the context of a Nida, not in the context of a zava (although on 69a there is a reference to a zava being hifrisha b'tahara). There is a machlokes tana'aim how early a Nida is able to be mafsik b'tahara. The issue is that during the 7 days there is an assumption that she will bleed and therefore, without a hefsek tahara she has no ability to change her status to an assumption that she isn't bleeding. The Mishna implies that the most lenient opinion (rabbonon) allow a hefsek tahara as early as day 2, but not on the day she started bleeding since she established herself to have a ma'ayan pa'suach (as the gemara explains in Rebbi's original position). In the end the the gemara implies that the conclusion of Rebbi is that she can even do the hefsek tahara on the first day.
When it comes to a Zav and Zava there is no mention of hefsek tahara, just a discussion about the minimum amount of bedikos. We pasken like R. Eliezer, but there is a machlokes Rav and Rav Chanina whether she must at least have bedikos on day 1 and 7, the bookends, or any one bedika in the 7 clean days is sufficient. Rashi (69a d.h. v'hacha) points out that the only reason we are going to allow the bedika on day 7 to retroactively serve as 7 clean days for the past 7 day is because she did a hefsek tahara prior to the start of the 7 clean days. The concept of doing a hefsek tahara before starting the 7 clean days is also indicated to by the gemara saying that a zava who is "hifrisha b'tahara" on the 3rd consecutive day of bleeding can count the next 7 days (starting from the next day) as the 7 clean days even though she doesn't do a bedika until day 7. It isn't clear though from the gemara how essential this is. Rashi takes it very seriously:
ומיהו הפרשה בטהרה בעינן דמשוחזק מעין פתוח ליכא לאחזוקי בטהרה עד שתבדוק ותמצא שפסק כדאמרינן במתניתין דהפרשה בטהרה בעינן
Rashi understands that the same requirement of hefsek tahara that we have for a Nida we have for a Zava, which implies that it is essential on a d'oraysa level. However, the gemara then assumes that according to Rav who holds that a bedika works on day 7, a woman can show up on day 7 and go to the mikva - we don't require ספורים בפנינו. Therefore the gemara asks from a braisa that indicates that we do require ספורים בפנינו on Rav who says you don't [Similarly, Tosafos (d.h. v'hacha) seems to take for granted that the hefsek tahara is essential according to Rav and therefore questions why the Mishna speaks about a bedika on day 1 since anyway we are speaking when there was a hefsek tahara with a bedika on day 7]. The Maharsha asks, what is the question on Rav, even Rav agrees that without a hefsek tahara it cannot count as 7 clean days, and she didn't do a hefsek tahara?
The Tosafos HaRosh (to which the Maharsha had no access) asks the same question. The Tosafos HaRosh suggests that the hefsek tahara for Zava may only be a d'rabonon, not a d'oraysa. Therefore, although a woman wouldn't be permitted to her husband without doing a hefsek tahara, she would be required to be to'vel at the end of 7 days by doing a bedika on day 7 (according to Rav who doesn't require ספורים בפנינו), if we assume טבילה בזמנה מצוה. The halacha of טבילה בזמנה מצוה would require that we are machmir to assume that perhaps she stopped bleeding in a way where the 7 clean days were effective in the absence of a hefsek tahara. Tosafos HaRosh then adds that even if we consider the hefsek tahara to be essential on a d'oraysa level, as rashi implies, we still must be concerned that maybe she did it and didn't remember and therefore we require her to be tovel on the day she comes by doing a bedika and considering it the 7the day of her 7 clean days.