Let's say I have a perfectly balanced dipole with a feedpoint impedance of
72+j0 at my frequency of interest.
I connect a perfect 72 ohm coax (an unbalanced connection) to the feedpoint,
and drive the antenna system with an unbalanced 72 ohm source.

Since the power sent to the antenna is not reflected, do I need a balun?

I think you're oversimplying things a bit here.

Under these conditions, without a balun, the coax feeding the
antenna is not going to see a perfect 72-ohm match. It will see the
72-ohm load of the antenna, in parallel with an unbalanced load
consisting of the conductive path down the outside of the feedline
coax towards the transmitter. Without a balun, the current flow into
the antenna will not be perfectly balanced, the feedline and load
impedances will not be perfectly matched, and there will be some
amount of reflected power.

Remember - current doesn't just flow in the *best* path. It flows in
*all* paths (unequally, depending on the impedances involved). The
mere fact that there's a "perfect match" between feedline and antenna,
does not somehow magically prevent current from flowing in other paths
which are attached to that same feedpoint.

Now... do you "need" a balun under these conditions? It depends, on
what "need" means.

- If somebody is going to shoot you if there's any measurable amount
of reflected power, or if there's any current flow on the outside
of the coax disturbing your antenna's theoretically-perfect
pattern, then yes, you need a balun.

- If your feedline happens to be close to a multiple of 1/2
wavelength long (as measured at the propagation velocity on the
*outside*, which will be close to C) between the antenna and
ground, then there may be enough current flow on the outside of the
coax to be problematic ("rf in the shack", or "rf pickup into the
house wiring or gutter downspouts"), and a balun would help choke
off the current flow and could be beneficial.

- If the feedline is long enough, and/or far enough away from an
even-halfwave length, then the impedance along the outside of the
coax will be high and/or reactive enough to result in a low level
of current flow. Under these conditions a balun may not make any
meaningful difference.

A lot of people run "half-wave" center-fed dipoles without baluns, and
are perfectly happy with the result.