I think health is fine as a general tag. It is quite useful to combine it with a species-specific tag (such as cats) when searching, and the number of results you get for combined searches like that seem pretty manageable.
While looking at some of the questions currently tagged health, I feel that the tag is maybe being used a bit too liberally.
It seems like in many cases, health is being used as a filler tag for anything that might positively or negatively impact the health of a pet. For concerned pet owners, that could conceivably apply to every question on the site, which does eliminate the utility of the tag.
The implicit idea that we all care about keeping our pets healthy means that health is redundant to questions like these:
Instead, I believe the tag should be used for specific health concerns or problems, such as:
The problem may be tied to the current tag wiki and excerpt:
The overall physical well-being of a pet. Personal medical questions are off-topic.
Includes tips for maintaining high levels of health for pets, and may cover such topics as diet, exercise, hygiene or dental-care.
This is too broad, and should be narrowed in scope.
I propose that the tag wiki excerpt be changed to:
Specific problems or concerns regarding the physical well-being of a pet.
I also propose that the tag wiki be changed (based upon this meta answer) to:
Addressing specific problems or health concerns that may negatively impact the well-being of a pet. Requests for general medical information are appropriate; however, personal medical advice is not.
Examples of appropriate general medical information questions:
What are the advantages/disadvantages of these three different treatments commonly used for $disease?
How often do indoor-only pets contract FIV?
$professional_publication had information about a promising new treatment for $condition a year ago; what further work has been done?
We suspect food allergies; what alternatives are there to the expensive prescription diet?
What is the general prognosis for $diagnosis?
Examples of off-topic personal medical advice:
Should I subject my elderly pet to surgery for $problem?
Should I vaccinate my indoor pet for FIV?
If I accidentally skip a dose of $medicine should I just skip it or double up the next one?
(Many personal details here, ending with) what should I do?
Additionally, the existing use of the tag should be reviewed, and questions that only vaguely touch upon the overall health of the pet should have the tag removed, leaving only questions where there is a concern about a specific negative health issue.