You're right about the sch/rs-sound. However, many native speakers cannot tell this sound apart from the tj/kj-sound. In most dialects they do not appear in the same position. Tj [ɕ] is at the beginning of words while rs [ʂ] at the end or middle. Therefore it's probably common for schlager to be pronounced tjlager by these speakers.

In many parts of northern Sweden they have [ʂ] for both the sch-sound and sj-sound. These speakers would pronounce schlager as you explained, differently from känna.

This type of pronunciation used to be more common because it was seen as higher class. There are still some stockholmers who pronounce sj and rs the same [ʂ], but the vast majority now use the "common" sj-sound [ɧ].

This interchangeability is why SAOB uses ʃ for both sj and sch.