Probably for the same reason exposure times are inconsistently rounded. The difference between the "actual" target f-numbers as powers of √2 and the rounded numbers we use is so trivial as to be beyond the limits of the vast majority of cameras in existence to accurately differentiate.

Why would anyone care if the rounded numbers used to describe the powers of the square root of two were imprecise to a lesser degree than cameras could actually achieve? Between variations in purity and age of chemicals, temperature control when developing and printing, as well as the limitations of manufacturing tolerances at the time, even half-stop accuracy was a pipe dream when several different systems for describing entrance pupil size, including using f-numbers, were being created. The rounded numbers were still more accurate than the capabilities of cameras to actually execute them, much less the more mathematically precise target values. Eventually the f-number system we all use today was standardized as alternate systems fell by the wayside. But even today the actual apertures achieved by even high end cameras and lenses are less precise than the inconsistently rounded numbers we use.

In other words, when we tell the camera to use f/22, it targets f/(√2)^9 (or approximately f/22.627), but where it actually lands could be anywhere from, say, f/21.5 to f/23.7. This is still within the range of 1/6-stop accuracy (roughly f/21.3574 - f/23.9729 for "f/22") that has been the general expectation for even high-end interchangeable lens cameras used to take artistic, documentary, historical, etc. photos since the latter part of the 20th century.1, 2

5.6 is twice 2.8, just as (√2)^5 is twice (√2)^3.

22 is twice 11, just as (√2)^9 is twice (√2)^7.

90 is twice 45, just as (√2)^13 is twice (√2)^11.

The numbers are just approximations of the actual targeted entrance pupil openings, where the camera/lens will attempt to make every full stop exactly half as large in area as the previous one.

When we tell the camera to stop down to f/5.6, it aims for f/(√2)^5.
When we tell the camera to stop down to f/11, it aims for f/(√2)^7.
When we tell the camera to stop down to f/22, it aims for f/(√2)^9.

Just as:

When we tell the camera to expose for 1/30 it aims for 1/32 (or 1/2^5 which is also 2^-5).
When we tell the camera to expose for 1/125 it aims for 1/128 (or 1/2^7 also 2^-7).
When we tell the camera to expose for 15" it aims for 16 seconds (or 2^4).
When we tell the camera to expose for 30" it aims for 32 seconds (or 2^5).

(You can easily test those last two with a stopwatch. Some cameras actually target the rounded value of 30 seconds, but many cameras target the actual power of 2 of 32 seconds.)

There are some f-numbers that are rounded to the same integer for some 1/3-stop scales and some 1/2-stop scales, but they're not the same target apertures with both the 1/3-stop scale and the 1/2-stop scale. For example, f/13 on the one-half stop scale (f/13.454) is slightly narrower than f/13 on the one-third stop scale (f/12.699). This is because the 1/2-stop scale is based on the fourth root of two, while the 1/3-stop scale is based on the sixth root of two. The 1/4-stop scale is based on the 8th root of two. 1/6-stop graduations are based on the 12th root of two.

¹ If you want more precise than that, you'll need to go with lab grade instruments that typically have fixed aperture openings. If you want to change the f-number, you must swap out the aperture diaphragm and then precisely align it in the center of the lens' optical path, rather than opening or closing an iris type diaphragm with multiple blades. For precision of exposure times, you'll control it using the duration of very precisely controlled lighting in an otherwise darkened enclosure, rather than by using a focal plane shutter.

² There are reports scattered about that most lenses with electronically controlled apertures are calibrated in 1/8-stop steps. Whichever 1/8-stop value is closest to the 1/3-stop value selected by the photographer is the one that is targeted when 1/3-stop intervals are selected by the camera's user. When 1/2-stop intervals are selected, the 1/8-stop calibrations line up, with every fourth step being either a full or half stop value.