Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

Those functions are much less used than before for one reason: the advent of electronic computers.

Before that, one had to rely either on tables or on slide rules. Tables were usually table of logarithms, and they included the logarithms of trigonometric functions as well. The trigonometric functions were then useful not only for geometric applications, but also to simplify algebraic calculations with logarithms.

For instance, to compute $\log\sqrt{a^2+b^2}$ when $\log a$ and $\log b$ are known, you could find $\theta$ such that $\log\tan\theta=\log\frac ba=\log b-\log a$, then $\log\sqrt{a^2+b^2}=\log a+\log\sqrt{1+\tan^2\theta}$ and $\log\sqrt{1+\tan^2\theta}=\log\frac{1}{\cos \theta}=-\log\cos\theta$. There are many similar formulas.

For geometric applications, sometimes versine and similar functions allow computing with greater precision while not adding too much computation. See for instance the haversine formula used to compute great circle distance (useful in navigation). The straightforward formula with arccosine has poor accuracy when the angle is small (the most common case), due to the fact that cosine is flat at $0$. However the formula with haversine is more accurate. To achieve the same, you would have to use $\sin^2(\theta/2)$ everywhere, which require more computations (but it's still reasonable with logarithms). Therefore, navigation table like Nories's nautical tables have an haversine table.

Note that tables of logarithms are usually accurate up to 5 digits (some larger tables had 7 digits, some very special ones had better precision but were difficult to use : more digits = much more space on paper). Slide rules have roughly 3 digits of precision.

All of this is rendered pretty useless with calculators, which have usually around 15 digits of precision and compute fast enough that we don't have to worry about speeding things up with extra functions.

Historically, those oddball functions were used primarily in navigation to reduce sextant readings and times to latitude and longitude. With the advent of radio, the need for that was greatly reduced, as we have radio direction finding, loran, and, more recently, GPS.