It's true, given the absence of anything like the Lott-Daschle
deal when this Senate was organized, that there wouldn't be an
automatic shift in control as there was following Jim Jeffords's
party switch. It's also possible that the Democrats would, through
the filibuster and other tactics, succeed in blocking any change in
control entirely.
But if Lieberman switched, the Republicans almost certainly
would advance a new slate of committee chairs to try to take back
control. At least in theory they could prevail. A major reason the
Senate didn't change control during the 83rd Congress was that
there was an independent voting with the Republicans to organize
the Senate plus Richard Nixon's tie-breaking vote. Deaths continued
to shift the partisan balance, but by that time LBJ had gained
effective control of the legislative agenda as minority leader.
Senate "Majority" Leader William Knowland's biggest accomplishment
during that period was the confirmation of Earl Warren.
I am very skeptical that Lieberman will actually switch, so I
don't expect that my theory is going to be tested anytime soon.