Your argument might have more weight if you could point to a process abuse akin to "deeming" a bill as being passed without actually voting on it, especially if that bill is signed into law as required by the Senate parliamentarian but the Senate never manages to pass the reconciliation bill (the only one the House would actually vote on).

In the history of the Republic I can't remember anything even close to this.

If you don’t remember it happening before, it’s because nobody has made this much noise about it before. When Newt Gingrich was Speaker of the House, for example, this very process was used 90 times. See the link I posted in my other comment.

"tu quoque" (party A did this so that makes it right for party B) is not a valid argument. If it was unconstitutional then, it's still unconstitutional now (even if it's now a tradition). (Cf. Gephardt Rule)

A process of "deeming" a bill passed without actually voting on it?

As I noted in my reply, "self-executing" is a red herring.