Welcome, willt9494!
I hadn't heard before "half-life" nor "parent-daughter" for nuclides widely separated on a decay chain, but I don't know everything neither.
What is clear is that the first disintegration, from
238U, is the slowest by far, so the subsequent ones influence very little the pace at which
238U disappears. Since
206Pb is the only end product, it's also the pace at which it's created.
The chain is detailed there, especially in a diagram
https://en.wikipedia.org/wiki/Decay_chain#Uranium_serieshttps://upload.wikimedia.org/wikipedia/commons/4/4e/Uranium_series.gifYour explanation with 50%, 75% and 25%... is correct.
If computing the products in small amounts after a short time, starting from purified
238U, the disintegrations with shorter decay times would have an influence too. The half-life from
234U to
230Th is a quarter million years, so one year after purifying
238U, the amount of
230Th would be limited not only by the decay from
238U but also from
234U.
By the way, detectors of ionizing rays catch essentially single particles, so they are damn sensitive, and as they distinguish the energy carried by gamma rays, they can also pick very low proportions of one radioactive nuclide among other ones. So together with mass spectroscopes, they can detect traces of a nuclide that would be impractical by most other means. Then the proportions of very rare nuclides in a decay chain can be significant. Very rare nuclides, and the ability to measure their proportion, makes also dating possible.
If comparing the rays emitted by nuclides in a chain, rather than the proportions of the nuclides, nuclides with a shorter live, which disappear quickly hence are rare, are also more radioactive - this cancels out: at "equilibrium" composition in a chain, as many disintegrations create and destroy an intermediate nuclide.