Is your question materially focussed on the hospital element? Or are you really interested in the age group element, and it just so happens that the hospitals are providing a sampling framework?
[EDIT: after writing this answer, I just saw the comment by whuber about "Grouping hospitals is tantamount to an assumption of homogeneity that ought to be checked." which is what I'm suggesting you address here.]
If the latter is the case, then you might consider taking an approach that is based around having clustered data, wherein the results from individual clusters (hospitals) are then combined to come up with an "overall" pattern of results.
Before I waffle on about clustering, given the huge number of hospitals you have these following methods may not majorly influence the results obtained compared to ignoring hospital (pooling results across all hospitals by age group, and then analysing as though all from a single sample.) This approach would in part depend on the relative sizes of the different hospitals (e.g. if there is a hospital serving a massive population that has quite different practices to other hospitals, then this hospital will dominate the results obtained).
Conceptually you could think of this as being a meta-analysis of the age -> treatment relationship, where each hospital is treated as a unique "study".
In practice, one could:
run some kind of multilevel model for this,
use tools designed for analysing clustered survey data (and specifying hospital as a cluster unit, with the regression model then looking at the age -> treatment decision relationship,)
calculate summary statistics within each cluster (hospital) -- for example, odds ratios between age groups -- and then use the mean and standard deviation of these odds ratios across all 800 hospitals to construct a confidence interval for the odds ratios. One could similarly calculate proportions per age group per hospital, then use similar methods to get some descriptive statistics around the proportions.
The following link (from Martin Bland) talks about randomised controlled trials, but most of the principles still apply to observational data:
Two outstanding points: You'll probably still run into problems with zero-counts or sparse cells in each hospital/age-group combination if your actual data look like the example (you will have odds ratios comparing groups on a per-hospital basis that are zero or infinity). There is also a second issue around treating the given/contraindicated/refused as independent outcomes -- I might have to pass on addressing this part for the moment.