Coriolis force and conservation of angular momentum
This is indeed confusing. The confusion comes from this very peculiar hypothesis:
What if the person doesn’t apply a tangential friction force at his feet?
It implies there is a radial contact force at the person's feet (I prefer "contact" to "friction", which refers to movement). And, indeed, for the person to move radially inwards, or even to stay immobile in the carousel, they need to at least counterbalance centrifugal acceleration.
So let's imagine how the person could be "frictionless" tangentially yet "frictionful" radially: suppose there are slippery concentric rails all over the carousel, on which the person can lean to move radially, but which prevent them to control rotational speed.
Suppose the person starts immobile with respect to the rail of radius $r$ on which they stand. When the person steps inwards, they undergo the said tangential Coriolis acceleration, which makes them start to glide counterclockwise along the inner rail of radius $r-δr$ on which they now stand, at $δω$ with respect to the carousel. Their rotational speed with respect to the lab is now $ω+δω$, and $δω$ is such that their angular moment has not changed: $rω=(r-δr)(ω+δω)$.