Chris,

Imagine the front forks of a bicycle. Now imagine two sets of forks, one
slightly smaller than the other so that it fits snugly inside. Now make
the column slightly narrower than the smaller set of forks and drill a
hole through it. Attach one set of forks to the tension member, the
other set to the roller support I described, then pass a pin through
both pairs of forks and the column. This joint would behave as I
explained, and does not in itself make the structure statically
indeterminate. Instead it makes it easier to analyse.

By its very complexity, what are describing is far more trouble than a
conventional pinned frame, which BTW is not difficult to analyze. In
fact, it can probably be done in less time than it would take to
correctly draw your proposed system.

In the words of Professor Hoover, "take it to pieces". If I understand
you correctly, the beam is not horizonally supported at the top, and is
therefore not in equilibrium. Add another horizontal support to fix
that, and it is no longer statically determinate.

Bill