Not true. Players on 2 way contracts still have to go through waivers if they are not waiver exempt. A 2 way contract only means the player makes different salary depending on where they're playing, typically more in the NHL and less in the AHL, ECHL, et al.
Brunnstrum was signed to a 1 year, 2-way contract. So technically he doesn't have to clear waivers to be called up. Only 1-way contracts go through the waiver process.
NHL Waiver Rules
Do 1-Way and 2-Way contracts affect Waivers?
Simply put, no they do not. This is a common misconception. 1-way and 2-way contracts have nothing to do with waivers. They are for salary purposes between the AHL and NHL. The confusion is understandable, most players with 2-way contracts are waiver exempt because of the reasons above (in the linked article). Most players on 1-way contracts aren’t waiver exempt. Just keep in mind that the chart above is the only thing that determines a players waiver status.
Plus, the Wings had to place Brunner on waivers to send him down to GR in October.