This mechanism would be prone to abuse: a group that opposes a particular issue could effectively time that issue out to prevent a replacement group from pursuing it.
The adage about “being the worst system except for all the other ones” applies well here: your (and my) participation in democracy is a required component, not just a nice thing to have. Devising hacks around our participation will ultimately result in a system that doesn’t allow our participation.
The mechanism is for people to vote for representatives who don't bring up those issues every year or two. The problem is that people just vote along party lines, and then vote for incumbents within those party lines. Without the threat of losing their office, politicians have no incentive to change their behavior from year to year.
Think about how your suggestion would work for other topics, like crime, immigration, social services, etc. It would undermine the government's ability to respond quickly to changes in the external world. It's not worth doing that over this one issue, when the real solution is to just vote for better candidates.
