the House could send clean spending bills for different agencies one by one up for a vote
First off, Trump refused to sign the continuing resolution (clean omnibus) bill passed by both the House and Senate before Christmas.
First order of business when the new House was seated was to vote for and send to the Senate the exact same bill. Which McConnell refused to bring up for a vote in the Senate.
Then the House immediately did what you said, breaking it up, passing each, and sending to the Senate where McConnell has continued to refuse to bring anything to the floor for a vote.
Trump, live on TV from the Oval Office, declared himself proud to call the shutdown his own and told Schumer to his face he would not blame the Democrats. So Trump owned the first part, and McConnell owns the second, running interference for Trump.
"For my part, whatever anguish of spirit it may cost, I am willing to know the whole truth; to know the worst and provide for it." - Patrick Henry
The level of injustice and wrong you endure is directly determined by how much you quietly submit to. Even to the point of extinction.