Unrelated to the article in question, but using ℝ over 𝕽 for the reals is more of a modern development. If you read older articles and textbooks, many will use 𝕽 rather than the sleeker ℝ (most in my experience, but results will be heavily affected by your cut-off for 'old').
I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.
I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.