This feels like one of those things where you're saying something that is maybe technically correct, so you can feel smart, but has no bearing on reality.

A store near me sells capers in a short-and-fat glass jar (which even a large spoon can fit in easily).

I've been able to leave those in the fridge just fine for 6 months with no noticeable change in quality... just like the tall-and-narrow glass jars.

Based on my personal experience of these two jar shapes, any difference in spoilage-rate is imperceptible.

What are you basing your "spoil faster" on?

I doubt the surface area matters. The volume of air and therefore oxygen you introduce every time you open the jar is the important factor. I would have thought that oxygen will dissolve in the liquid over a day or so even with a small surface area between the air and the liquid.