"Take the role" vs. "take over the role" vs. "take on the role"

There is no significant definitional difference between the three expressions, but they differ in connotation, so they cannot exactly be used interchangeably either.

"To take a role" is a neutral expression, which can be used in a variety of ways but usually serves as a denotation that implies nothing else. It doesn't seem like what you're looking for.

"To take over a role" is, as you said, to take a role that was previously occupied by somebody else, which seems to be what you're looking for, as your colleague "replaced" another worker, so he "took over" that worker's role.

"To take on a role" is, again as you said, to accept the role as a challenge. However, you seem to be emphasizing the replacement aspect more than the challenge aspect, at least from what I can tell by your description.

I'd personally recommend "take over the role".