Quotation marks at the start of all the lines of a paragraph?

That's indeed a good pointer. The only two missing details were the box-decoration-break: clone; and that it actually can work only on line (not block) boxes:

blockquote {
  font-size: 20px;
  line-height: 1.25;
  padding-block: 1em;
  background-color: #fffdca;
  color: #6b644a;
  text-indent: 10px;
}
blockquote span {
  box-decoration-break: clone;
  background-image: radial-gradient(closest-side, currentcolor 0 100%, transparent 0);
  background-repeat: no-repeat;
  background-position: .5em center;
  background-size: .3em .3em;
  padding-inline: 1em;
}

They have their Temples, whereof the chief, as the Jewish Temple did, stands in their metropolis; and is named Almack’s, a word of uncertain etymology. They worship principally by night; and have their Highpriests and Highpriestesses, who, however, do not continue for life. The rites, by some supposed to be of the Menadic sort, or perhaps with an Eleusinian or Cabiric character, are held strictly secret. Nor are Sacred Books wanting to the Sect; these they call Fashionable Novels: however, the Canon is not completed, and some are canonical and others not.

Sample adjustments: the blockquote content is wrapped in extra span (necessary, sadly) that is decorated with dots (for simplicity and independence from external resources).

Correct, there is no "hatch" implemented for that, be it through generated content or some other property. There is some discussion going on about dusting off the old ::line-marker idea in csswg-drafts/issues/13329 (which points back here to SO to Alohci's demo using box-decoration-break: clone, applied cleverly to semantic inline , what seems like a good idea even for your use case perhaps).

Emulating historical typesetting in "clean" HTML + CSS is hard; take for example the beautiful extended whitespaces between sentences featured in your example: not possible without extra markup. Or granular control over automatic hyphenation: some browsers have some support for some languages, but overall still not perfect. Part of the problem possibly is, that such typesetting actually took many manual interventions adjusting the outcome, that ought to be reflected in a similarly manual fashion in the code. But that's probably a different story.

This quickly fails on different screen sizes and line widths…

Indeed. Testing it right here, using Chrome 145, it didn’t reproduce the quote mark when wrapping added additional lines. In looking for possible solutions, I did find a CSS Trick from 2018 that might provide some ideas to experiment with, depending on how flexible text-decoration might have subsequently become. I will note that I have never actually seen the usage that the querent is seeking to reproduce; within the last fifty or so years, the “standard” seems to be that in a multi-paragraph quotation, only the first line of each paragraph bears a quotation mark.

So there's really no built-in way is there? I thought there might be something similar to the text-emphasis property, except for lines, but not dependent on the line count.