The framers drafted the constitutional provisions against the backdrop of impeachment as it had been practiced in England, where the House of Commons impeached and the House of Lords tried the impeachments. The whole point of impeachment by the Commons was for the charges of impeachment to be brought against the accused in the House of Lords.
Strictly speaking, “impeachment” occurred – and occurs — when the articles of impeachment are presented to the Senate for trial. And at that point, the Senate is obliged by the Constitution to hold a trial.
What would make that trial fair is a separate question, one that deserves its own discussion. But we can say with some confidence that only the Senate is empowered to judge the fairness of its own trial – that’s what the “sole power to try all impeachments” means.
If the House votes to “impeach” but doesn’t send the articles to the Senate or send impeachment managers there to carry its message, it hasn’t directly violated the text of the Constitution. But the House would be acting against the implicit logic of the Constitution’s description of impeachment.