What are the historical reasons for the conventional sequence of footnote symbols?

According to @Mahnax's answer to this question, the Chicago Manual of Style Online states that the correct sequence of footnote symbols is as follows:

* (asterisk; but do not use if p values occur in the table; see 3.78)
† (dagger)
‡ (double dagger)
§ (section mark)
|| (parallels)
# (number sign, or pound)

What were the actual reasons that gave rise to this sequence? Could it be traceable back to the typewriters of 50 or 100 years ago having keys in that order?

This Wikipedia article on the dagger (archived by Swarthmore College Computer Society) claims:

History

The symbol was first used in liturgical books of the Roman Catholic Church, marking a minor intermediate pause in the chanting of Psalm verses (the major intermediate pause was marked with an asterisk) or the point at which the chanting of the Psalm was taken up after an introductory antiphon whose words were identical to the opening words of the Psalm.

Usage

The dagger is usually used to indicate a footnote, in the same way an asterisk is. However, the dagger is only used for a second footnote when an asterisk is already used. A third footnote employs the double dagger. Additional footnotes are somewhat inconsistent and represented by a variety of symbols, e.g., parallels (||) and the pilcrow (¶), some of which were nonexistent in early modern typography. Partly because of this, superscript numerals have increasingly been used in modern literature in the place of these symbols, especially when several footnotes are required. Some texts use asterisks and daggers alongside superscripts, using the former for per-page footnotes and the latter for endnotes.

Robert Bringhurst's 2005 The Elements of Typographic Style (version 3.1) (nicknamed Bringhurst's Bible by typographers) says the traditional order of symbols is *, †, ‡, §, ‖, ¶ and goes on to say:

“But beyond the ... double dagger, this order is not familiar to most readers, and never was.”

The order doesn't come from typewriters, which were invented in 1868, because they were used as footnotes before this. Here's an example from a 1792 magazine:

And the asterisk, dagger, double-dagger, section mark:

But they also used numbered footnotes, especially when there were many footnotes, such as these nine:

Here's Blackwood's Edinburgh Magazine from 1819 with six footnotes; asterisk, dagger, double-dagger, section mark, parallels and pilcrow:

The asterisk is used before a dagger to denote birth and death dates. Typographers Hoefler & Frere-Jones say:

Both characters have functions in genealogy and other life sciences, where the asterisk indicates the year of birth (*1499), and the dagger the year of death (†1561). There are standard fourth-, fifth- and sixth-order reference marks, too: they are the section mark (§), parallels (||), and number sign (#), after which the cycle repeats with doubles, triples, and so on: *, †, ‡, §, ||, #, **, ††, ‡‡, §§, ||||, ###, ***, †††, ‡‡‡, etc. Beyond three, numbered footnotes are always preferable, even if you are David Foster Wallace.

The Sweet Sound of Punctuation by Yves Peters says of the dagger:

Because its shape is reminiscent of a Christian cross, in predominantly Christian regions the mark may also appear before or after the name of a deceased person, or the date of death. Therefore, it's not used as a footnote mark next to the name of a living person.

It also says another name for dagger is obelisk. So: asterisk and obelisk => Asterix and Obelix!