If you define R as "the average number of people an infected person goes on to infect", then looking at the 1000 cases in the community who infect 800 others in in the community and 400 others in hospital, they clearly have R=1.2 (the sum of Rc and Rch). So it's not surprising that the overall R (for cases in hospital and community) is greater than 1.

By partitioning R into multiple components, you create a false impression: each individual partition is less than 1, but the sum is greater than one, and that's why the overall R exceeds 1.

If you take this further and partition R into R(home), R(outside), R(at work), R(travelling), R(hospital), R(carehome), etc. and all the cross-infection rates, you could get each individual component below 0.1 (say) and still have an overall R much greater than 1.

It's simply misleading to compare a particular partitioned value of R with the overall R without making clear which is being referred to.