Your cycling fallacy is…
“People cycling are allowed to share bus lanes, which are wide and fast, so there's no need for cycling-specific infrastructure”
People who already have the confidence to cycle amongst heavy motor traffic may find bus lanes to be a suitable environment in which to ride. Indeed, for those people, bus lanes can offer many benefits over a general traffic lane.
But for the vast majority who don't fit this narrow demographic, sharing space with large vehicles such as buses is deeply unappealing. Additionally, because of their size and mass, buses also represent genuine danger to people cycling.
Furthermore, the stop-start nature of bus journeys isn't compatible with the constant speed of cycling, which results in a 'leap-frog' situation, where the person cycling overtakes the bus at stops, meaning the bus has to try and overtake between the stops. In other words, buses hold up cycling, and vice versa.
If large numbers of the population were to take up cycling in bus lanes, the bus system would become unviable, as buses would be delayed due to too many people cycling – so cycling in bus lanes is not a realistic long-term solution for cycling.
Recent research also suggests that bus lanes offer no safety benefit for people cycling over roads that have no bus lanes at all.
Source: As Easy as Riding a Bike (Copyright, used with permission)
Source: As Easy as Riding a Bike (Public domain)