i read it as closing closing rather than closing for repair too.
Indeed, even on re-reading it there is not even a suggestion that a repair or diversion may be diversion - just a panicked demand for closure.
Of course the local authority's rights of way team may well just LEAP on this demand with glee. It will save money, time, effort and aggravation to merely close off the bridleway - and could even result in a bit of 'you pat my back, I'll pat yours' from any landowner across whose land the bridleway passes.
Whatever, it is likely to result of the disappearance by default of yet another link in the already-fragmented chain of rights of way across this land, the disappearance of living history and the stories that it carries and shrink the boundaries of our freedoms even more. It may well be 'just one bridleway' in an area where there is a generous network of bridleways - but that statement applied, once (and not so very long ago, either), to a great many places in England and Wales ...
All that I ask is that people think through their contacts with RIghts of Way authorities before making requests or demands which may impinge on the freedom of future generations to access our countryside by non-motorised means.
Clearly a bridleway which is boggy to the extent described by the OP is dangerous. Many things in life are dangerous, but that does not mean that they should be closed or forbidden. The danger is often - as in this case - clearly identifiable and able to be mitigated or avoided.
It is doubtful that the original path would have passed through a bog of such depth, so the historical route of the path may need to be checked, as should the existence and upkeep of drainage ditches and field drains; perhaps a spring or a stream has been either naturally or artificially blocked or diverted, or earth has been dumped into or excavated from an area which affects the drainage of the bridleway. Closure of the bridleway will have no effect on any of these things, and may indeed merely encourage rogue landowners or neighbours to try their own means of making a bridleway dangerously impassible, in the hopes that it, too, will be officially closed by the local authority rights of way team ...