Many of his contemporaries hated his guts and ultimately they condemned him to death. It may be a good way to keep an open mind, but it may also be a terrible way to make friends and influence people.
He wasn't condemned to death because of his annoying questions, he was condemned because he was considered partly responsible for the "Thirty Tyrants" rule of terror which replaced democracy in Athens. Socrates was vehemently opposed to the democracy, and one of the leaders of the tyrants had been his student. When democracy was restored, Socrates was condemned to death.
I think it's because the political process is all about getting people into groups where they can thus cooperate with each other as an economic bloc, a perfect lie is the optimal unifying concept in this model; something that pulls everybody together into the same single political structure.
If on the other hand all you care about is the truth, you're directly corrosive to the above. You're looking at all the edge cases where the lie breaks down, and worse yet you're spreading disintegration of the otherwise unified political bloc by infecting other agents with your same methods.
It's a matter of perspective which side is "right", because those resultant atomised and fractured political blocs that can no longer bring themselves to accept the beautiful lie that otherwise would have successfully united them are now competitive rather than cooperative, and the game gradually slips closer to zero sum with the bloc most closely pursuing the optimal strategy in the light of the cold hard cynical truth winning out at the direct expense of all the other groups, and the resulting accelerating wealth inequality that implies, having real concrete negative effects on the lives of all those people in the suboptimal factions that frankly they may never have even had a chance to join letalone have been made aware of the existence of any alternatives because of the nature of their worldviews. Is it "right" to pursue truth even if it makes the quality of life of billions much worse?
It's a frightening and enlightening thing to sit down with an ideologue and come to understand not just what they think, but how they got to think that way. The common thread I have found is that default worldviews are both extremely sticky and subject to almost no critical analysis by the people that hold them, and unifying the galaxy of irrational but widely held default worldviews that exist flatly requires extensive narrative manipulation and outright lying, and that lying and manipulation is what politics actually is.
Imho this is why widespread censorship has gone from intolerable anathema to the sine qua non for the existence of the dominant shared mass hallucination about the state of the world in just a few short decades. Like it or not, politics has won out soundly over truth past a certain social scale.
Disclaimer; acknowledging reality is not approving of it. Socrates was right and should have beeen feeding his prosecutors hemlock, not the other way around. Damn the consequences and embrace the truth, whatever the outcome has always been my view. I just also know that view is extremely unpopular today.
Many of his contemporaries hated his guts and ultimately they condemned him to death. It may be a good way to keep an open mind, but it may also be a terrible way to make friends and influence people.