why can't I search for clients who need sitters and etc?

Because that is not the way Rover has chosen to operate. When owners post "Help Wanted" type ads, they're gong to be inundated with responses from sitters or walkers. While it may be more work for owners to read sitters' profiles and contact several before meeting and making a decision, it is ultimately better for the customer not to be barraged by zillions of responses, many of which may not even be suitable.

As Walt has written, there are or were other sites that operate as you described. But I do understand why Rover has elected not to go that route.

Bottom line is that option isn't available. The entire system would have to be changed to support this, what you with to have is the Wag model where the client can post a walking request.