What would anyone who doesn't care enough do? If they don't vote, they increase the power of other voters. If they vote randomly, their votes would cancel each other out on average, making it as if they hadn't voted. If they vote based on what somebody or an organisation says, that person/group is empowered based on the number of followers.
This is the main issue with FPTP voting. If you wish to disempower fringes, require majority approval.
It only empowers the fringes if the rest of the general public does nothing. One would assume that people would start caring when the fanatic fringes start influencing public policy. If they still don't care, then they forfeit any right to complain.
You can start caring all you want, but if you have to have an informed opinion on every vote, you are going to run out of time, even if you do not have to spend time to do such things as bringing in the money to live from.
Also, who would propose the votes, check legislation for consistency and loopholes, etc? If you crowd-source that, chances are 100% that there will be some decisions with unwelcome consequences. If you appoint people for doing that, how is that different from letting those people make the decisions, too?
There would still be legislatures who's job is to propose and write laws, but the voting and passing of those laws could easily be shifted to the people instead of the same people that are writing the laws. This would further balance the power, and prevent the extreme amount of lobbying and special interest groups from controlling our laws.
I don't expect anybody to vote on all laws. It makes more sense for people to vote on the laws that are relevant/important to them. If I don't know anything about a certain issue, I shouldn't be voting on it anyway. This way, people would vote on issues relevant to them, and for people who don't care, then well, they don't care...