Yes it varies unpredictably based on changes in the distribution of mass within the Earth from magma flows, air currents, ice pack movements, ocean currents, and so on.

https://en.wikipedia.org/wiki/Day_length_fluctuations

More like the precision of ± 0.5 seconds / year, that is to say 15 parts per billion, is that ludicrous. For comparison, time acceleration in geostationary orbit due to general relativity is somewhat less than one part per billion. Household tools (other than clocks) rarely go beyond a percent, and if you need a particular quantity you don’t have a ready-made tool for you’ll likely going to be hard-pressed to go beyond ten percent.

The more precise you want to be, the more complex and numerous physical phenomena you need to consider become; with a system as involved as the Earth, at some point you get to weather-like chaotic behaviour, so no practical amount of additional precision in input data helps anymore.

Nearly all things in nature spite our systems.

The earth even speeds up and slows down in response to stuff like earthquakes and volcanoes.

It doesn't help that the atomic second was defined using old data such that when it was defined we were already relatively far off from 86400 seconds/day. A relatively large part of the correction being applied by leap seconds comes from this initial offset, which is also why all leap seconds so far have been in one direction. (and why the handling of the other direction is essentially completely untested, making the prospects of a negative leap second in near-ish future pretty frightening).

Basically, things like the weather can influence the speed of rotation. Angular momentum changes when mass is closer to the center of gravity or farther out. That measurably includes mass like water in clouds and leafs on trees.

It would surprise me if we could not measure and detect any variation in the earth’s rotation. A leap second is on the order of less than 0.000001% variation, at least according to the leap seconds applied recently.