Why shouldn't we bring back lynchings?

The government dishes out the death penalty all the time but when a community comes together to lynch a criminal everyone throws a hissy fit.

I thought this was supposed to be a democracy.

On the face of it, lynchings do look democratic enough; some slick demagogue convinces people that you need to die, usually to pay for an unsolved murder case or something. Object all you want, but the people have voted.

The actual problem is that those people haven't been given much of a trial-worthy case. More often than not, the demagogue in question has been feeding the crowd some emotional BS, filled with half-truths or even outright lies. And the emotional frenzy he's whipped up, makes it outright dangerous to start poking holes in his …logic.

Much better, then, the controlled and orderly forms that a courtroom presents¹. There people can demand evidence with something like impunity, and it is now up to the demagogue in question to actually supply a case based on logic, reasoning – and, well, evidence. Actually prove that you really did what he says you did, have logic and reasoning dictate what fate you really deserve.

Greatly reduces¹ those hideous miscarriages of justice, that lynchings actually invite by their very essence.
 

¹ On paper.
Doesn't really take the risk away completely, just reduces them significantly. But there's still those cases where some dickhead puts his prestige and authority into making it a show trial. Which is, among other things, why we need constraints on who gets to conduct cases. Nobody who just simply personally hates you, nobody who loves your money/woman/job/whatever and needs you gone, nobody that needs to be seen as «not soft on crime» for some upcoming election, etc.
And, obviously no MAFIAA members that will not make products worthy of your money, now that they see a way to make more money through protection-money schemes like LawsuiTerrorism.

Tl;dr: Court-rooms are, in theory, much safer for the innocent than are lynchings. That's been eroded away by show-trials over the centuries, but that's actually somewhat of an easier problem to address.
And, hopefully, fix.