i didnt go through all these pots so i dont know if this particular point was addressed, but its quite clear that one has a mitzvah min hatorah to rid chametz from his home even on pesach. there is a discussion in the rishonim as to whether or not the mitzvah of tashbisu is to be in a chametz-free state when chatzos comes, or if it means that when chatzos comes we must now rid ourselves of chametz. however, the rambam who clearly paskens the former understanding, says that if someone does a bedika on pesach because he forgot or was unable to do so before pesach, he makes a bracha on the bedika, whereas after pesach he would not (and yes, he would need to do a bedika after pesach. i cannot recall offhand the particular halachah but its either the second or third perek of hilchos chametz umatzah). the least it would be would be a mitzvah to remove the possibility of being over the issur of bal yeraeh ubal yimatzeh, but truthfully it seems quite clear that a person has a positive mitzvah to remove the chametz from his reshus on pesach as well.
as for gebroks, i enjoy my matzah brye and mtzah balls every pesach, and plan on continuin to do so. i cannot understrand the minhag, it makes absolutely no sense. the shulchan aruch brings lihalachah that a zaken or choleh who cannot chew and swallow the matzah can soak it in water and be mekayem his mitzvas aseh of achilas matzah with said matzah (provided that it doesnt become completely broken up because of the water). there is also the halacha that matzah can never bcome chametz. i cannot grasp the notion that the same matzah that one will eat to fulfill his tora obligation of matzah, will at the same time be suspected of not even being matzah. is there not a more tarti desatri than that? anyways, enough of my rant against gebroks. the reality is, 99% of people who keep it keep it because its their minhag. just like kitniyos. everyone would abandon such a minhag in a heartbeat if they had the ability to, but what can we do, i guess we need a ??? ??? ????? ????? ??????