In my point of view, although it is very important to have a measure of research/researcher scientific quality, sometimes I feel that h-index needs to consider other parameters rather than citations only, such as publishing in a review journal or specialized scientific journal.
It is a good index, provided its taken in context. In particular it should be interpreted for the career stage. A score of 15 for a 28-year-old is impressive, but less so for a 58-year-old. For extra insight it is also good to look at how many of an academic's publications are 'first author'. Put less value on an h-index if the academic isn't the first author on any or many of the publications. The h-index gives an idea of the quality of the actual academic output of individuals. It doesn't really matter how good an academic's qualifications are, or how many publications they have, or how high the impact factors of the journals are, if they are not getting their work cited. The h-index tells you that: is the person's work being cited or not? And that's the important thing. Considering it's just a single number, it's quite informative.
As far as age of researchers is concerned, we certainly agree that very distinguished professors, who may publish in tough journals such as JFM (Journal of Fluid Mechanics) for example,a may have small h-index. Some Journals need specific readers, particularly those of deeply scientific theories and physical interpretation of findings.
Journal of Fluid Mechanics has an impact factor of 3.2, therefore, on average a professor who published a paper in that journal ten years ago would have about 32 citations today. If that was there only publication that the professor managed in their career, they would have an h-index of 1. However, If they published about 30 papers of that quality they would have an h-index of about 30. The h-index works, but to get a high figure you need to publish lots of different research that is well cited. The editor of Journal of Fluid Mechanics has an h-index of 39. He is the the editor because he is an accomplished researcher, which is reflected in his high h-index.
It does take the quality of work into consideration. That's the point of it. If you just count the number of publications somebody has, that number does NOT indicate anything about quality. The benefit of the h-index is that it does contain information about the quality of the work, via the number of citations.
I think in some occasions, many citations may not be due to the deep analysis or methodology a paper introduces, but rather because the manuscript is a review article. Hence, this in the point of view of some readers would save time and effort if reading original papers, and accordingly may rise the number of citations.