Wood working (carpentry? Whatever.) is not exact.

Woodworking is the trade of building furniture and cabinetry, carpentry is the trade of building structures. A woodworker built your dining room table, a carpenter built your dining room. The two trades have some overlap in skills and knowledge but they are different skill sets. As for exactness...let me put it this way. A lot of woodworkers balk at using pencils to mark their cuts because the width of a 0.5mm mechanical pencil lead is considered too great a margin for error. A single bevel marking knife is used to apply a score mark with a perfectly sharp edge to the wood.

Dividing 19mm by 3 is a weird example.

It is typical for metric woodworkers to mill boards to a finished thickness of 19mm. It's also a common thickness for plywood in metric land. I think it was chosen because it is close to 3/4". Many common woodworking joints such as mortise and tenons, bridle joints, etc. require dividing the board into thirds; a typical mortise for example is a rectangular hole or slot in a board 1/3rd the board's thickness. So working in metric with 19mm stock, you either have to cope with measuring, marking, and cutting 6.3333mm, or having to "just know" to cut a 7mm mortise with 6mm of wood on either side.

Meanwhile working in US Customary (which is not the same as Imperial) using wood milled to 3/4" stock, a third of 3/4" is 1/4".

19mm/3 isn't a weird example, I didn't pull that number out of thin air.

DIFFERENT SCENARIO: Now I'm going to make a half-lap joint which requires cutting the stock in half. Cutting a metric 19mm board in half gets you 9.5mm. Cutting a US Customary 3/4" thick board in half gets you 3/8".

For this application, fractions genuinely don't suck. There are advantages to using fractions in work like this, namely that you can do integer math on the numerator or demoninator rather than floating point arithmetic. Plus, measuring and marking tools being marked in powers of two rather than ten is more convenient in a field where most of what you're doing is halving and quartering dimensions.

Sure, precision metalwork is done in thousandths of an inch or even ten-thousandths of an inch, and I personally prefer machining in metric.

A 2x4 is not a 2x4.

Yes it is, for awhile at least. The board is rough sawn out of the log at 2" by 4" and dried at that dimension. Rough sawing doesn't produce a perfect board, and the board will shrink and warp a little during drying, so the dried board is then further flattened, straightened and squared via a milling process which takes about a quarter inch from each face, resulting in a finished dimension of 1 1/2" by 3 1/2". Lumber is priced by their rough cut dimensions because that's how much of the tree the sawyer had to use to make that board.

Back in the day it was common for lumber yards to sell construction lumber in a rough cut state at a true 2" by 4", and the carpenter would mill it himself. Then the railroads happened, and lumber was being shipped from the forests of the West coast back east. Railroads charged for cargo by the ton, and lumber mills could save a mint on shipping by milling the boards to finished size before shipping. This saved carpenters the work of milling the boards themselves. They still called the boards "2x4s" because they were still used for the same purpose. An thus the modern commodity retail 2x4 was born.

Similarly, that 3/4" lumber I keep saying I use: I buy that from my local sawyer rough sawn to 1" thick. I then plane it flat and straight, which takes about an eighth of an inch from each face. So I wind up with a finished board 3/4" thick, which as previously discussed is a convenient size for woodworking.