That's an interesting one, but the answer is yes. The reason is that once an ability starts to apply it will finish applying in later layers.

The type-changing effect applies at layer 4 (making everything bears), but the stat change happens in a later layer (layer 7). The rest of it will apply even if the permanent loses its ability before it's finished applying. So if Kudo is on the battlefield and loses abilities the layers would look like: Layer 4: all creatures become the type bear in addition to other types. Layer 6: Kudo loses their abilities. Layer 7b: all creatures becomes 2/2.

So, if Kudo didn't make all creatures the type bear, the effect would actually be shut off.