The trouble with your example is that each person within the community infects 1.2 others (0.8 in community + 0.4 in hospital); each person in hospital infects 0.9 others.
But your argument is sound. A more striking example: within a nodal group ( could be hospitals, could be supermarket staff) each infected person infects 5 others within that group, 5 in the general community; in that general community each person infects 0.01 in the nodal group, 0.08 in the general community - a really tight lockdown. Assume the nodal group is just 1% of the total community and initiate with 1 infected person in the nodal group, 99 in the general community. After 7 cycles you have 95000 infected in the nodal group, 96000 in the general community. Overall R rapidly converges to about 5.01 .
After 11 cycles the number of infected in the nodal group has reached the entire population of the UK - which is nonsense. What happens of course is that the nodal group develops herd immunity, and the infection dies away. This explains why a tight lockdown takes so long to show results. From that point, everything is OK, until another nodal group which has been suppressed under lockdown goes back to work!