Should ordinal indicators be inline?

No, neither choice is a universally accepted convention, and a reviewer is more likely to know the chosen convention for the journal or academic setting in which this discussion is occurring. You might get the style sheet for the journal (it probably being available as a pdf file) and do a search for 'superscript'.

It has always annoyed me that Microsoft Word has so many style conventions that are imposed by default. When I get a new installation I need to go through and rip out about 50% of their "helpful" style auto-substitutions.

Proper style with ordinal numbers is a matter of taste between publications.

It may be helpful to remember that indicating ordinals with arabic numerals was viewed, until perhaps the end of the 19th century, as somewhat gauche even if unavoidable. The more preferred procedure was to write out the ordinal as a word, as we still do on formal invitations. (Of course, with the scientific and industrial revolutions ordinal numbers higher than "thirty-first" became much more common in use.)

In my early days as a printer, I always used "in-line" ordinals because they were easier to set in type. The necessary six superior characters (s t n d r h) were usually not available on Linotype, and even when they were available they had to be inserted, slowly, by hand.

This is the origin of the common U.S. scholarly style of in-line ordinal indicators: practicality for the Linotype, which was the primary typesetting machine in use by U.S. printers until the late 20th century.

A similar situation prevailed with typists, who would need to resort to minor gymnastics to insert superior characters: so they would base-align superiors if they could, often inserting a hyphen before them.

In years prior to the introduction of the Linotype, house styles would vary, depending on availability of handset superior type characters.

United States typefounders did not usually cast as wide a range of characters in their type founts as did typefounders in any other country. Because English requires a bare minimum of letters, with no accent marks being strictly necessary, a monolingual American job printer, or even a book printer, had little need for "extra" "sorts".

Thus, the majority U.S. practice for printers and typists for well over one hundred years has been to use base-aligned regular-sized letters for termination of ordinal numerals, although the use of superiors has always had its exotic appeal.

Because much book typesetting in the U.K. was predominantly on the Monotype machine, use of superiors was not usually viewed as such a great hindrance, but it still required devoting six character positions in the caster to essentially "quirky" letters. Six character positions (per font) would make it impossible to assign other special characters that might be of more practical value, such as accented characters, foreign monetary symbols, or scientific and mathematical symbols.

Because most Romance languages use only two different superior letters for indicating an ordinal (either o or a), typesetters in countries using Romance languages could use these two superiors as standard items, especially in view of the common practice of abbreviating "Numero" as "N" with a superior "o". Consequently, European use of superior "o" or "a" was normal, especially in view of the possibility of confusion of a base-aligning lower-case "o" with a zero.

What proper English language style should be in the days of computer typesetting is, in my view, a matter of taste. From a production standpoint I would prefer in-line ordinals because they do not interrupt the keyboarding with formatting instructions, and thus also do not cause formatting errors that must then be corrected.

I do not find the introduction of a hyphen to be a bad idea: "the 22-nd Amendment". I doubt it would ever become a standard, however, despite its clarity in distinguishing ordinals from possible typographic errors ("Did he mean "first" by "1st", or was it a typo for "last"?) Insertion of a hyphen is common in mathematical copy where, for example, "to the x-th power" reads much more smoothly than "to the x th power".