No, I do not believe that a few thousand people - of greatly varied ages, physical condition, intelligence, intent, etc. - COULD take over the US Government.
However, the then-sitting President, one of his sons, one of his lawyers, and various enthralled Congress-critters and other minions did wrong on January 6.
They summoned the mob... they whipped-up the mob... they aimed the mob... they loosed the mob... they observed its progress for hours before acting.
The whole point of the mob was to halt the formal certification of the results of the Electoral College vote connected with the election of November 3 2020.
The mob succeeded in delaying the formal certification but the mob could not cancel it altogether and therefore ultimately failed in its designed mission.
Halting the formal certification of the voting results of the Electoral College would have kept the previous incumbent in office until the matter was settled.
Giving the incumbent more time to conjure-up solutions ( seizure of ballot boxes, declaring the election invalid, martial law, whatever ) to keep him in power.
Keeping him in power would have had the effect of overthrowing the US Government and the Constitution in a philosophical if not a physical sense.
Had it succeeded this would have been what is known as a "bloodless coup" (nearly bloodless, anyway).
As it is... charges of conspiracy, sedition, insurrection and defrauding of the US government will have to do, in order to punish and to discourage a repeat.
Given that the Chairman of the Joint Chiefs of Staff had issued a caution to the White House about Martial Law and Lawful Orders. Rump had no recourse.
His summoning and activation of the mob - mostly through his lieutenants [ ol' Teflon Don ] - was a last desperate measure to try to hold onto power.
It was an arrogant, selfish, disloyal, un-American and traitorous act that will go down in the history books as a grotesque and indefensible infamy.
Rightly so.