Not necessarily. Two major schools of Tibetan Buddhism do not operate this way: the largest, Gelugpa and as well as Sakya.
You are on thin ice here as both Gelug and Sakya recognize and train tulkus. Neither are completely devoid of tulkus although the relationship with the recognition can be different. You in fact note this wrt Sakya sub-sects. Your example wrt the office of the Dalai Lama is strained at best. In fact Gelug tulkus can indeed be sought (other than the Dalai and Panchen lamas).
Most so called "Sakya" tulkus are Nyingma reincarnations trained in Sakya through circumstances of region. The Zimog tulkus of Nalendra are a notable exception to this. As I said most of the Sakya tulkus come from East Tibet, which is very removed from Central Tibet. Even here, Sakya in Derge belongs to Ngor, not Tshar, and Sakya monasteries in east Tibet were under the adminstration of Ngor. Of all four schools, Sakya has the least number of tulkus.
I certainly trust the Sakya lamas to be able to negotiate a repudiation of tulkus as an institution if they saw that the institution was not beneficial. BTW - I was startled when a Sakya lama said to me flat out that tulkus were nirmanakayas. There was absolutely no hedging of the issue on his part.
AFA Gelug is concerned — there may be an effort to find Tulkus, but the administration of Gelug is in the hands of educated geshes, not reincarnations, for the most part.
Wrt to both paragraphs, administration was not under discussion (at least not originally).
What are the purposes of tulkus? Well one of them *IS* to bring in money, esp. as a result of pilgrimage tours. However the primary purpose is to benefit beings in some way. This can be very open. Lama Lodro notes five basic purposes in his bardo teachings book (miraculous activities, meditation, scholarship, institution building [I'm pretty sure he has another term here but it's really institution building, and teaching - I'll have to double check this though). Even so these can be pretty broad categories and practically anything can be subsumed under these categories. Tulku Thondup goes into these activities and more in detail in Incarnation. The Red Book (history of the Nyingma, Dudjom Rinpoche) talks about the purposes of tulkus in several places.
The basic job of a tulku is to benefit beings. Very open-ended with few detailed position responsibilities (at least in some cases, officially - high reincarnations are definitely restricted and constrained from birth in many cases). We can say that the purpose of recognized reincarnations is to do something extraordinary and this is the primary hope and promise of the institution. The institution is a means of injecting charismatic activity into Tibetan Buddhism (one avenue for that and now the primary one since mahasiddhas generally don't exist much anymore and are not showing themselves openly). The institution can also be seen as a secondary means of lineage transmission, apparently wrt terma transmission (the story of Apong Terton's Red Tara beig transmitted to HHST fits this). In the west most tulkus were clearly expected to become teachers and this had mixed results.
_________________Kirt's Tibetan Translation Notes
"Only you can make your mind beautiful."
HH Chetsang Rinpoche