Please contact the developer of the router if you think that it's not working properly. OSM is about data and access tags are about legal restrictions.

Routing through carriageway should not be forbidden for those who want to use it. It's up to the people to decide where it's safer. Someone may not want to be near kids at 40+ km/h or reduce his speed to a pedestrian speed at every crossing (required by law) or maybe taking the curbs aren't really comfortable on his bike. If a car fits on the road then a bike fits too. There is no such thing as a "car road" in road traffic act.

For example, brouter has different profiles to accommodate different cyclists. It's up to router and the user to make the decision. See how it takes bicycle-pedestrian path with Safety profile https://brouter.m11n.de/#map=14/59.5262/24.8634/standard,route-quality&lonlats=24.837491,59.515656;24.90476,59.504387&profile=safety

Personal example: I usually don't bike fast, but here https://www.openstreetmap.org/way/204921428 
when going downhill I always take the carriageway since I have quite a high speed and the sidepath has people walking, drivers pulling out from private houses not expecting anyone fast, there is also a non-see-through fence at the end restricting visibility before crossing which is dangerous if you're cycling fast, as drivers pull out from Metsarahva tee quite fast.