> For example, we don't really have a president.
This actually nothing to do with being a direct democracy. You still have an executive cabinet selected by (but not responsible to) parliament. It's just that every member of the executive is selected by the parliament, instead of the parliament selecting a leader who picks the rest.
> I get to vote on what I believe is right on every issue, instead of getting tricked by politicians
You can still get tricked by politicians. America has very many citizens-initiated referenda, but it cannot be said to have improved their democratic outcomes.