I would agree it was just Democrats shouting in the dark... if there didn't keep being problems. See here's the thing: The issue isn't with the e-mail leaks. That's not what is being talked about, it is if any of Trump's associates had illegal ties to the Russians and more importantly if Trump tried to cover it up.
Trump was told that Flynn was likely compromised and he shouldn't hire him. Had he not, well that story would end there. But he did hire him. He then pressured the FBI director to drop the investigation in to Flynn, and only fired Flynn when it leaked that he had this conflict of interest. He then implied the problem, and the reason he fired Flynn, was the leak not the compromise. Then he later fired the FBI director which his people claimed was related to the e-mails but he came out and said was because of the Russia investigation.
Guess what? That shit starts to look a lot like obstruction of justice. That's why this thing continues to have legs.
Oh an impeachment of a president? That's not "corrupt politicians" "overthrow[ing] of an elected President," it is constitutional, and is what is supposed to happen if the president breaks the law. Article one section 1 states that "The House of Representatives shall choose their speaker and other officers; and shall have the sole power of impeachment." Article two section four states "The President, Vice President and all civil officers of the United States, shall be removed from office on impeachment for, and conviction of, treason, bribery, or other high crimes and misdemeanors."
So ya, if it turns out he obstructed justice, which you'd need an investigation to determine (and that is what is going on), accepted bribes, or other illegal acts then the House would be within its constitutional power to impeach him and the Senate to try him. That's not some covert scheme to subvert the Constitution, it is written right in the original text.