In his autobiography, Goethe recounts a conversation with his friend Klopstock about whether the German word for an ice skate should be Schlittschuh or Schrittschuh. They
spoke namely in good southern German of Schlittschuhen, which he did not accept as valid because the word does not come from Schlitten, as if one puts on little runners, but rather from Schreiten, that is, one, like the Homeric gods, strides over the sea become land on winged shoes.Trunz 1948, III.61–62. Translation my own.
The authors of Spuren auf dem Eise have a definite opinion on this:
In some parts of northern Germany, people say Schrittschuh following Klopstock, who discussed it with Goethe in a funny argument. Naturally Goethe was right: the ice skate is modeled on the sled, and the experienced skater does not stride but rather glides or rides and lifts the foot only as much as is unavoidable. Actual Schrittschuhe are just shoes.Diamantidi et al., 1892, 2. Translation my own.
Actually, they’re both right. Schlittschuh is the descendant of Schrittschuh, which was formed from schreiten back in the days when it meant glide instead of stride. Once the meaning of schreiten shifted, German speakers decided Schlitt- made more sense than Schritt- in an example of folk etymology. There’s a short summary and references in Fowler (2018, 81–83).
D. Diamantidi, C. von Korper, and M. Wirth. 1892. Spuren auf dem Eise: Die Entwicklung des Eislaufes auf der Bahn des Wiener Eislauf-Vereines. 2nd ed. Vienna: Alfred Hölder.
G. H. Fowler. 2018. On the Outside Edge: Being Diversions in the History of Skating. Edited by B. A. Thurber. Evanston, IL: Skating History Press.
E. Trunz, ed. 1948. Goethes Werke: Hamburger Ausgabe in 14 Bände. Hamburg: C. Wegner.