The Democrats are claiming that the mere impeachment without conviction of President Trump is evidence enough of insurrection to trigger the 14th Amendment and prohibit him from holding office.
The first problem with this theory is that an impeachment isn’t a conviction, but merely an accusation. This would hand the lower house of Congress the power to deny anyone to run for any Federal office, merely by voting to impeach them.
Even if this theory were true, how will they enforce it? I remember that there were allegations that Obama didn’t meet the Constitutional requirements for the Presidency, but the courts ruled time and again that no one had legal standing to challenge his candidacy or presidency.