Checks and balances means some folks should NOT be directly elected. if everyone is <directly elected>, then you have <directly elected> checked and balanced by <directly elected>. Which is to say, not at all. :-P
You could have a system where everyone is directly elected while keeping checks and balances, if voting were restricted, eg. maybe everyone can vote for a president/prime minister, but only non-teachers can vote for an education minister, and only non-finance people can vote for something like the Fed chief, etc. The point being the checks and balances now happen because other groups keep your group in check by voting.
one if the problems is that most elections are only for one person, so only the majority (the person with the most votes) wins.
give everyone half a dozen votes or more, and and you'll get a more representative sample.
for example instead of electing a president, elect a while leadership team. independent of party affiliation. (i'd get rid of parties completely while we are at it, every candidate should be independent (the expanded version of that gets even rid of candidates, every adult can potentially be elected, but that is a more complex system that needs more elaboration))