As to what would actually improve (read fix) the American democracy, I'm not sure a mandatory voting law would actually have much effect. While it would undoubtedly increase voter turnout, if it does not result in the election process internalizing the interests and preferences of the currently disengaged voter, then it would not alter the policy outcomes in any meaningful way.
In my experience people who are able to vote but don't, do so for one of three reasons: (1) no confidence in the system, (2) lack of history/culture of voting, or (3) lack of information. (1) is a whole mess in and of itself, which I won't expand on unless someone is interested.
(2) is perhaps the easiest to understand. Voting, and voting regularly are a learned behavior. Most people who vote in every election come from a family and/or social environment where that is the norm and the expectation. It is most apparently seen with the activity of the parents, if a child grows up seeing their parents vote in every election, that child is much more likely to vote and to vote regularly. This goes along with the general way in which parents greatly determine political affiliation as well. You can see this phenomenon most obviously among Black voters in the South. Within the Black community there exists both a segment which can be relied upon to vote regularly (not necessarily every election) and a segment which can be relied upon to vote rarely (not necessarily never voting). While some of this breakdown occurs based on economic status, typically those who do not vote are the ones who would most benefit from their preferred candidate winning and are most harmed by their least preferred candidate winning. The most salient difference between these two segments is the attitude of their peers and family. Those who don't vote tend to come from families that don't vote and live among others who do not vote. [Interesting note, some of this comes down to the Church they attend, as some Black Churches strongly encourage voting (not necessarily for a specific candidate or party) while others do not discuss voting in any meaningful/encouraging way.] In other words some non-participatory voters are non-participatory simply because that's their tradition, rather than for any rational reason. It also means that even if gains are made in any single election year, if they are not repeated consistently and continuously, then there will be no lasting change, and lasting change will only really be felt generationally.
(3) refers to people who do not know there is an election, do not know that the election affects them, do not know when/how to vote, or do not know that they can/are expected to vote. While this is the smallest group, and the easiest to correct, it will likely remain an issue that is not addressed as this set of people largely lean towards one political pole. This is in fact the largest hurdle to any voting reform in the U.S.
Generally speaking those who don't vote tend to prefer one party to the other, and it tends to be the same party in most geographic areas. While this support is not always present initially, upon becoming informed as to the platforms and policy stances of the parties the support tends to be majoritarian even if not overwhelming. As long as one party benefits from large sections of the populace not voting, they will oppose any proposal which makes it more likely that these people begin to vote.