A brokered convention would be a disaster for the DNC and probably completely shatter it for good. It's already so splintered, that would be the last straw and Democrats would probably not win a national election for decades.
What are Democratic voters going to do, vote for Republicans?
A brokered convention only hurts when the losers abandon the party. 1972 and 1964 being the most recent obvious examples, but also, to an extent 1984. (And the parties recovered
massively within two years in all three cases.)
Brokered conventions were the norm for decades and that was when the parties were (at least pretending to be) relative clones of each other.
1964 is an interesting comparison, Goldwater had a first ballot plurality going into the convention and he was almost guaranteed to win a majority from the favorite sons, they still tried to deny him the nomination at the convention. It couldn't be done. So the left-wing of the Republican Party just fucked off and let him get destroyed. Eight years later, there was nearly a similar situation in 1972 at the Democratic convention, McGovern had a plurality and nobody could come close so they tried introducing new candidates at the convention. In 1980 and 1984, many Democrats were hoping Carter and Mondale could be first ballot blocked and open up things to a second ballot that ignored the primary candidates. Robert C. Byrd was trying to position himself in 1980 to be the compromise choice between the Carter and Kennedy camps until Carter beat Kennedy in the primaries. 1984 is what caused the superdelegates as Mondale entered with a plurality with a likely majority, although Hart/Jackson had nowhere near enough to deny him totally they could have potentially deadlocked the convention on the first ballot so Mondale's team spent most of the convention trying to lock down a first ballot majority.
The fear isn't that Bernie voters do what they didn't in 2016 and stay home, it's that they do what they did in 2016 and try to start taking over the party. And this time will have even more allies, especially if the party were to go with Bloomberg/Buttigieg.
If Trump wins, we're going to have a billion candidates in 2024, and it'll be the start of a re-alignment of both parties. It's not just the Democrats, the GOP has to figure out if Trumpism will continue past Trump himself. For every Republican who has spent the last 40 years claiming Reagan's mantle, that's come with tons of infighting over whether or not they deserve it. Yet none of them had Reagan's ability to reach outside the party to protect his flanks. Trump has a similar, but not as powerful, makeup of his base. A lot of Republicans hope once Trump passes they can go back to how it was. Same as Democrats hoping they can just wait out Bernie. Except it's never really been Trump and Bernie specifically as much as how they fit right into existing fissures in the parties. Trump most notably on immigration, trade, blue collar workers, the large anti-war base he's intellectually tied to, etc. Bernie touches a lot of similar issues in the Democratic Party where a large part of the base is not happy with the elites positions. I'm not sure there's anything so stark as immigration was for Trump to where Trump could spend a couple months being the only GOP candidate to actually run on the party base's majority position of anti-immigration as everyone else demurred then hastily tried to fake it. Remember "self-deportation" and other stupid straddling? But there's similar stuff, the Medicare For All waffling and defenses of private health insurance as the glory of humanity are pretty silly. The positions on Obamacare at times are hilarious. It was a compromise of a compromise at the time but you'd think it was now Social Security after 70 years or something.
