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So if I want to take a Knight alongside some...other, greyer knights, what exactly am I losing from the Knight side of things?

I bring a Super Heavy Aux with one unit.
The Knight detachment rule says, "An IMPERIAL KNIGHTS Detachment is one that only includes models with the IMPERIAL KNIGHTS keyword (excluding models with the AGENT OF THE IMPERIUM or UNALIGNED keyword)."
The Wandering Hero rules says, "If this Detachment is a Super-heavy Auxiliary Detachment that contains one FREEBLADE unit, until the end of the battle, that unit gains the AGENT OF THE IMPERIUM keyword. Only one FREEBLADE unit in your army can have this keyword."

So my Super Heavy Aux is not a Knights Detachment, is that correct? Which means no strats, no relics, and no counting as extra models for objectives, no ALLEGIANCE, no HOUSEHOLD

Incorrect.

You must be an Imperial Knights detachment for the Wandering Hero rule to apply to your detachment. When it say excluding Agents of the Imperium, it means despite the existence of such models in the detachment, not to ignore those models entirely.

Thus per a Super-heavy Auxiliary Detachment that contains one Freeblade Imperial Knight unit, then it is an Imperial Knights detachment for which all the normal detachment rules apply (despite being a SHAD) and that unit also has the Agent of the Imperium keyword. This may cause special interaction with other rules, but does not in and of itself do anything.

You benefit from being a freeblade

E.g. hunter of beasts

That's it

Your Freeblade wouldn't benefit from the usual Imperial Knights purity bonus, but it won't stop your other, greyer knights from getting theirs.