If I were a Democrat in Connecticut, I'd vote for Ned Lamont. Having said that, Lieberman has every right in the world to run for his party's nomination, and, losing that, preserve the option of running as an independent in November. If its "democracy" to challenge a sitting incumbent within a primary, there should be no objections to that same incumbent trying to win by appealing to voters outside the party. Elections are won by getting more votes than the other guys, not having the election handed to you in a one-party state because you win your primary. If Lieberman believes he would win a plurality of votes by running as an independent, bully for him.
I just don't want to hear about any sitting Democratic Senator (that means you, Charles Schumer) knifing Lamont in the back if he wins.