Often they are hundreds of years old, so any new construction needs to allow the public to cross over them. When I was growing up, one of my neighbours did a lot of research into footpaths in the area, and uncovered a lot of forgotten footpaths. Some of them even went through people's back gardens, and as they were legal rights of way, they had no choice but to open them up (apparently there's a way to move them now for cases like this now).
These bridges are needed because a legally mandated footpath crosses a railway line. That's why so they're lightweight - because they're often very rarely used - they're there mainly for legal compliance.
You can't cycle on these kinds of footpath - both legally but also literally because they're usually just mud and across agricultural fields and you would have to cross stiles. You couldn't cycle to them even if you wanted to.
So no there is no cyclebridge, because nobody would be trying to cycle over them.
First you berate someone for talking about bicycles because "footbridge" is in the name, then you ask how people would get to "this bridge", which makes no sense because there was no specific bridge in the first place, then when asked what bridge you are talking about you jump to "cultural context".
The context is that nobody would want to bicycle over these bridges in general because where these bridges are generally used it is not legally or physically possible to get to them on bicycle.
The point therefore is your question doesn't make any sense if you know the context.
Look at the other comments saying the same thing as me if you want another perspective to help you understand if you're struggling.
I don’t know if you’re a non-native speaker? ‘This bridge’ means ‘this bridge design’.