What we do is take a concept (in this example, telling the time) and look at how we can use it to apply other areas of knowledge. So we would learn to tell the time, but also use the clock face when we look at fractions, or times tables and so on.
The only indispensable reason to teach a child how to read an analogue clock is to give a child a feel for the passage of time, and even this is a fruitless task since a feel for this is partly a function of physical brain development.
You didn't mention geometry as a reason to teach clock reading, but it is actually the only mathematically relevant area for using the clock face, though even there it requires adaptation for complete relevance. Each minute = 6 degrees, the circle has 360 degrees, angles are represented visually by the two hands, and can be calculated, etc. This is the only area where the clock face functions within a system. In every other area it is an adjunct, a detour, and an example of various concepts with limited applicability.
There are many better ways to directly approach fractions, five times tables, and the concept of numbers simultaneously holding varying values.
Cuisenaire rods and other manipulatives that are specifically designed for their purpose convey those concepts much better. The clock face has distinct limitations in terms of relevance to clear thought, reason, and the wide world of numbers. It's a closed system operating with a numerical limit of 60.
We live in a base 10 world. At best, the clock as a teaching tool can reinforce other concepts in the base 10 system - five times tables, the fact that a whole can be divided into ever smaller parts (but even this has limited value and clocks are not the only way to show this - cake, pie, but above all actual math manipulatives work too). You are hoping to develop the faculty of logic here, to teach a general rule that can be applied, not teach tricks.
If you are trying to provide a foundation for future concepts, you need to make sure the example you are working with has no limitations and that your examples will be relevant within the base 10 system. Place value, the multiplication operation, and fractions are only partly shown by a clock. It's an everyday concept, not a scientific concept or a system.
As mathematical examples, cake and pie and other foods (and dividing up the clock face) refer to sharing or dividing - they present the idea of fractions as a partition. That's fine as far as it goes - it's a basic, everyday concept, but it has to be subsumed into the general system and enlarged in order to have relevance and move the student forward.
There are cognitive issues involved with irrational numbers, not just why the fractions are getting smaller but the numbers are getting bigger, but the idea of the whole as 1 vs a measurement of 1 unit within the whole. What is one third of a tape measuring 1.3m? What is one quarter of a tape measuring 1.3m? The meaning of the notation must be addressed. A tape measuring 1.3m with the 33cm increments marked and the 1m point highlighted would work better.
Teaching is only useful when it moves ahead of development.