I take it that, in the video, the drawers the woman put on were essentially white capris for modesty purposes?

In the 1870s, women were still wearing drawers (btw - we generally use "drawers" as "bloomers" has a specific meaning - in the 1850s, the bloomer costume is a gown with a shortened skirt and loose "Turkish trousers", and in the 1890s bloomers are full, loose pants popular for bicycling; "pantaloons" are an early 19th century form of men's trousers) with two separate legs sewn together only at the waistband - that is, fully open at the crotch. With these, it was not difficult to sit/crouch over a chamber pot without having to remove anything - servants not necessary.

These drawers first began to be worn by somewhat risque ladies around the turn of the century, but there was a great deal of societal disapproval due to the fact that, being bifurcated, it seemed short of like cross-dressing. Prior to this, women simply didn't wear anything on their legs at all - their underwear consisted of their shift and stays (chemise and corset, in 19th century parlance). While pantalettes - essentially, drawers so long they were visible - were fashionable for both make and female children in the 1830s-1840s, drawers don't seem to have come into general use for women until sometime after the invention and popularization of the hoop skirt in 1856: hoops had a tendency to bounce and swing and potentially reveal everything. But, you say, if the drawers were split, wouldn't they reveal everything as well? Au contraire! Women usually wore a very short petticoat under the hoop, and the back of the leg pieces are also nearly always overlapped.

At this moment, as far as I am aware, academics have not paid any attention to the "under or over chemise" debate and it remains entirely between reenactors. As such, discussions tend to come down to personal comfort, and so I cannot make a ruling on the subject at this time.