In their textbook you'll find that galaxy rotation curve studies are spectroscopic. That is they are keenly interested in the relative redshift in the 21 cm neutral atomic hydrogen lines (among others like H2 and CO molecular lines) at the limbs of edge-on disc galaxies, and adapting that to spirals and other disc galaxies that are tilted away from edge on. [Binney 2e sec 6.1]. Practically invariably the relevant lines are relatively redshifted and relatively blueshifted at the limbs, leading to the interpretation that generically there is equatorial spin in disc galaxies.
In elliptical galaxies there is practically no equatorial spin to speak of; instead the spread of relative redshift across the face of the elliptical is interpreted as blobs of hydrogen gas moving radially, that is sinking deeper into the galaxy or rising out of the galaxy's depths. It is also useful to study a wide range of absorption lines given that the clouds are backlit by a galaxy's worth of starlight (and sometimes a quasar), in a process which grinds out surface densities.
These relative redshifts do not depend on cosmological redshift (the whole galaxy, or its whole cluster, is cosmologically redshifted identically for all practical purposes, so the opposite limbs in discs are affected similarly). It is also not sensitive to an isolated galaxy's peculiar motion within a cluster. It may matter for merging galaxies.
We can also look deeper than disc limbs and ellptical surfaces. Optical interferometry is highly sensitive in this application, and provides direct evidence of the motion of the various sources of emission and absorption lines from various gas clouds, dusts, and even starlight. The Large Binocular Telescope does some work in this area. Radio interferometry is useful for looking into the bulk motions within the more central regions of galaxies and clusters; dust obscures optical signals but millimetre signals cut through.
The investigated starlight is bulk and while the interpretation depends on assumptions about the bulk stellar chemistry of an observational target (maybe metallicity varies slightly in different parts of an elliptical which might have a history of galaxy mergers) there is in no way a dependence upon any single star and its meanderings through its galaxy. We're interested in the light generated by ~billions of stars, not the positions or momenta of single stars, mostly because we just cannot resolve the latter with current technology.
> as an outsider
You seem interested in the topic. Learning how observations are made (and the history of them) is probably not inaccessible to you given your comments here. Whether that leads you into any sort of conclusions about the structure and evolution of galaxies is up to you, but I think textbooks will be better for you than whatever ultimately led you to the stackexchange link in your comment. (I did notice however that the Ciotti preprint discussed later in your link cites Binney & Tremaine multiple times, as does the Ludwig paper in the stackexchange question).
