Why do we still use a 24/60 based time system?

Humanity has had the need to measure things for thousands of years, most importantly because of commercial reasons. Managing large amounts of produce, storing it, transporting it, selling it, buying it and so on becomes really complicated and confusing if there is no way to measure and express quantities. Knowing how much of what you have is extremely useful. Selling produce likewise becomes so much more manageable when prices can be set based on units of measurement (after all, not everything can be sold at a given price per piece, because "pieces" can be too small for this to be feasible, like grains of wheat or rice, or basically impossible to be divided into "units", like liquids.)

Unsurprisingly, over the millenia different peoples came up with different ways to measure stuff, using different units of measurement. Since there was generally very little contact between distant lands, isolated and semi-isolated peoples came up with their own units independently of each other. In addition, measuring and measurement units were heavily driven by custom and tradition, which meant that different things would often use different units of measurement, even though they all measured the same thing (such as weight or volume).

Indeed, for the longest time you could go to eg. an open-air marketplace and encounter a dozen completely different measurement units of weight, used to measure different things (eg. flour could use its own measurement units, wheat could use its own, peas could use their own and so on), and a dozen different units of volume (eg. beer could use different units from wine, which could be different from units of water) and so on.

The Imperial System of measurements succeeded in somewhat standardizing, unifying and replacing all these hundreds and hundreds of different other measurement systems, but many hold-outs and relics still persisted to some extent, and co-existed alongside the Imperial system for quite a long time.

At some point the scientific community got tired of this mess. As science progressed more and more, and measurements became more and more important and universal (and especially the accuracy of measurements became important), a single unified standard was needed. The Imperial system was awkward to use because converting between the different units (eg. inches, feet, yards, miles...) was difficult. On top of that, science was in more and more need for extremely small and extremely large units of measurement, for which the Imperial system was rather inadequate for.

They needed a simpler, easier to use, and more consistent measurement system that also allowed easily going arbitrarily small and arbitrarily large, so they came up with the metric system, which eventually became the standard in science. While the base units in the metric system are rather arbitrary, the conversion between different scales of the units is 10-based and thus very trivial to do. This better system replaced all of the thousands of previous systems (with the exception of the Imperial system in some places, which still refuses to die).

But one has to wonder: Given that the metric system replaces and defines all units of measurement, using a 10-based system... why doesn't it also do so with time? Why are we still using the ancient 24-hour, 60-minute, 60-second time measurement system? Fractions of a second are done using base 10, but other than that we are still stuck with this odd system that makes conversions harder than they could be (eg. how many seconds is 5.21 hours? Not easy at all to do in your head.)

The main reason for this is that by the time when the pressure to create a single unified measurement system was large enough that it was done, time units were already pretty much universally standardized.

While you could find a dozen different measurement systems being used even at the same marketplace, every single person there was nevertheless using the same hours, the same minutes, the same seconds. Nobody was using any other units of time. This form of timekeeping had somehow become universal, and had been so for very long time. Thus there was no impetus to change it. It was just easier to keep the system that everybody was already using anyway.

(It's not like nobody tried. Some countries did try to create a 10-based timekeeping system, with 10-hour days and 10-day weeks. All these attempts failed miserably because everybody simply ignored it. It never caught on.)

So in the same way as the Imperial system still exists today and is widely used in some countries, the 24-hour 60-minute system likewise still exists today and is still ubiquitous. Tradition. There was never any impetus to change it, so it was never changed.