It has been the general American practice regarding criminal law to grant considerable discretion to prosecutors, so that by analogy one could argue that the House has complete discretion to decide whether to initiate impeachment proceedings. On the other hand, Alexander Hamilton, in The Federalist No. 77, argued that the nation would find “republican” safety from a presidential abuse of power by the mode of his election and by his “being at all times liable to impeachment.” There is no doubt that the Framers saw impeachment as a part of the system of checks and balances to maintain the separation of powers and the republican form of government. The implication is that when the President (or other impeachable official) has committed an impeachable offense, the Members of the House, bound by the oaths they take to uphold the Constitution, are under a particular obligation to deal with the miscreant’s offenses, irrespective of whether their bill of impeachment may or may not lead to a conviction in the Senate.
Surely, this is why a Republican-led House initiated impeachment proceedings on Bill Clinton. Why should the current Democrat-led House not be bound to the same obligation to hold proceedings on President Trump.