Is a hefsek tahara required for a zav/zava?

In nida on 69a, Rav said that a zav or zava doesn't really need a bedika on the first day and on the seventh day but one or the other is enough. He says techilasan af al gav she'ein sofan and sofan af al gav she'ein techilasan. However, when he says those halachos he actually says it about two specific cases. One case is that the bedika was done on the first and eighth day and Rav says that is ok even though there is no end. The Rishonim (including Rashi) point out that Rav's halacha is true even without the bedika on the eighth day. All you need is a bedika on the first day and that is enough. Why then did the gemara set up the case where there was a bedika on the eighth day? Just to show that even in that case R' Chanina argues.

The gemara then talks about a woman who did a hefsek on the last day of her ziva bleeding. She then did not do a bedika on the first day of her 7 nekiim and did a bedika only on the last day. In that case Rav says that's ok as long as you did the bedika on the seventh then you don't need a bedika in the beginning. Tosafos explains that according to Rav that bedika on the last day of bleeding is necessary. Just a bedika at the end with nothing at the beginning or before the beginning is nothing. You need the hefsek tahara and the only question is if you also need a bedika on the first clean day. To that Rav says it's not necessary.

I have a number of problems with this explanation (although I didn't see anyone who argues with Tosafos and I think it has to be true).
1. The two statements of Rav really mean different things. When he says techilasan af al gav she'ein sofan, he means that you don't need any bedika at the end at all (even after the end) but when he says sofan af al gav she'ein techilasan he really means that you don't need one at the beginning but you need one before the beginning. (This question doesn't bother me that much but it's just strange that the two statements aren't parallel.)
2. The gemara explains that Rav had to make both statements because if there was no bedika at the beginning is a much bigger chidush than if there was no bedika at the end. The gemara says that is because even if there is no bedika at the end at least there is the chezkas tahara because of the bedika at the beginning but if there is no bedika in the beginning then there is no chazaka. According to Tosafos though there was a bedika before the beginning - the hefsek tahara! So why is that any worse and why is this a bigger chidush??
3. Tosafos Harash asks if the hefsek is required then what was the gemara's question on Rav from the braisa. The gemara tried to prove that Rav was wrong because when the confused woman doesn't know if she was a yoledes b'zov we tell her that in the first week she doesn't need to go to the mikva during the day because she first needs to count 7 nekiim. This shouldn't have been a question because maybe she didn't even do the hefsek so that's why even according to Rav you need a bedika at the beginning.
Tosafos Harash answers this question that it could be that the hefsek is only d'rabanon but tevila b'zmana would override that. Therefore even if she didn't do a hefsek, according to Rav we would still require her to go to the mikva right away. He also suggests another answer that maybe she did a hefsek but she just doesn't remember.