There are axial/centrifugal flow engines around - the old Bristol Proteus was one such and I think some of the smaller French engines.The axial stages are in front of a final centrifugal stage - about the only advantage I can see is that the engine can be built shorter. Centrifugal compressors are very robust - I worked for about 13 years with RR Dart engines and never saw one pulled for compressor damage but have seen many axial flow engines with sometimes extreme damage from bird ingestion. FOD and even damage due to surge and stall. Centrifugal compressors work over a larger rpm range although they run at higher rpm -up to 15000rpm on RR Dart and a whopping 42000 rpm on the Garrett TPE 331. They do not need airflow control devices such as bleed valves or VGVs. An axial flow engine without stators would not work - you need both to get compression.