The purpose of the hefsek taharah, the daily bedikot during the seven clean days, and wearing white underwear, is to confirm that the bleeding has actually stopped. So yes, under normal circumstances we definitely want to know what is going on during the seven clean days.
On the other hand, in a case of need (such as when a woman is experiencing problematic staining, especially if this is a frequent occurrence), there is room to be lenient about disregarding stains, even during the clean days. The laws of stains are a rabbinic stringency added on to the Torah laws of niddah. Chazal did not intend for staining to unnecessarily burden women and cause them to constantly be niddah, so there are leniencies built into these laws. Thus, a flow of blood, or staining with hargashah, is certainly problematic on a Torah level and would interrupt the clean days. A flow of blood or a hargashah may not be ignored. But if the bleeding is staining rather than a flow, and not accompanied by hargashah, a woman may take advantage of the leniencies of ketamim (stains) in order to disregard the staining and complete the clean days.
While l'chatchila (ideally) one should perform two bedikot daily during the seven clean days, b'dieved (after the fact, or in a situation of significant need) one bedikah each on days 1 and 7 is sufficient. Therefore omitting the bedikot in the middle does not invalidate a set of clean days. A woman who experiences difficulty completing the clean days may take advantage of these leniencies and reduce the number of bedikot, or skip bedikot when she suspects she is staining. This is not a l'chatchila situation, but exists b'dieved to help those who need it.
There is no halachic requirement to check one's toilet paper, so even during the clean days one may avoid looking for stains on the toilet paper.
Please feel free to get back to us with any further questions.