WI: US Senators drawn from Representatives?

Imagine this slight tweak to the US Constitution: Adding a qualification for members of the US Senate that they must have served at least one full term in Congress (for all intents and purposes, the House).

Let us posit that this would not go into effect until at least the year 1800, either due to the structure of the clause or as an amendment passed around that time (otherwise, there'd be no qualified people to sit in the Senate until the second Congress, at which point, the Senate would cannibalize almost half the House). That way, when it goes into effect, there's a reasonably large pool of current and former Representatives from which to draw the Senators. I specified one full term in Congress rather than the House in the initial proposition so that sitting Senators would not be disqualified (and maybe it would be best to phrase it as 'one full term in the House of Representatives, or 2 full years in the Senate,' to account for Senators appointed to fill out terms, but we're getting in the weeds here).

What impact would this have on politics in the US?
 
I feel like this would be hard to work because of the existence of pesky 'ol at-large congressional districts.

Would it? First, a state could draw their Senators from other states' former/current Reps. Further, even if each Senator only served one term and they were picked from their state's former/current Reps, the rate at which Reps are being put into office is still faster than they'd be being pulled into the Senate.
 
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