" but do I need to?"

Yes.

The wizened one was talking about corrosion by dissolved oxygen. The
water in a sealed system system will typically absorb less oxygen than
that in an an open system, but it is likely that there will still be
some oxygen getting into the water. An inhibitor would contain an
oxygen scavenger, which reacts with the oxygen and renders it harmless
before it reacts with the inside surface of the radiators.

ALSO, because the system contains different metals (steel rads, copper
pipes, brass fittings, etc.) and an electrolyte, there will be
galvanic corrosion of the steel radiators. The inhibitor will contain
a pH adjuster which will make the water alkaline and so much less
conductive as an electrolyte, stopping the galvanic corrosion. If the
system hasn't been thoroughly flushed, it may contain acidic flux
residues which will accelerate the galvanic corrosion. The inhibitor
would neutralise any flux residues, within reason.