Why do party leaders prefer the convention system?
Party leaders prefer the convention system simply because it gives them more power over who their party nominates to run for a given political office. That makes them more important and gives them more influence over the people who hold governmental office.
In the convention system, candidates are chosen by some number of party activists. Even at a relatively large convention, the number of delegates is really very small when compared to the number of people in the overall population who are members of a given party. This means that, at a convention, candidates are chosen by something of an inner circle. This inner circle gets to pick the candidates that it likes.
Contrast that with our current situation in which party leaders have very little clout. For the most part, nominations for political office are made through primary elections. There are, of course, party conventions where the presidential candidates are officially nominated, but it has been decades since these have played any real role in selecting candidates. Instead, candidates who want to get nominated need to play to the voters, not to the inner circles of the party. This is why, for example, “Tea Party” candidates can get to run as Republicans even when the party leaders would have preferred more moderate candidates.
This is why party leaders would prefer conventions. Conventions would give them the power to decide who would get to run for various offices. This would give the party much more control over its message and might allow it to run what it thinks of as better candidates.