Why didn't "spiel" get spelled with an "sh"?

As others have mentioned, spiel may actually be derived from German Spiel rather than, or in addition to, Yiddish shpil.

In German, syllable-initial /ʃ/ (the "sh" sound) is written with the trigraph <sch> before <r>, <l>, <m> and <n> (i.e. letters that represent resonants), but with the single letter <s> before the letters <t> and <p> (i.e. letters that represent plosives). I think this explains why we don't see /ʃ/ represented by <s> in words such as schlep, schlock, schmooze. German words spelled with <st> and <sp> are generally anglicized in English pronunciation to have /st/ and /sp/ (e.g. see this question about Einstein).

Yiddish romanization is considerably less standardized than German orthography, but in general, /ʃ/ is represented by <sh> in all contexts, including syllable-initial <sht> <shp>. There are some variant spellings used in English that have <scht>, such as "schtick" (a variant spelling of shtick); I didn't find any commonly used word spelled with <schp>, but people certainly might use that for non-standardized transcriptions of Yiddish words.

Spiel derives from German spielen not from American Yiddish like the other terms you mention (boldface mine):

• "glib speech, pitch," 1896, probably from verb (1894) meaning "to speak in a glib manner," earlier "to play circus music" (1870, in a German-American context), from German spielen "to play," from Old High German spilon (cognate with Old English spilian "to play"). The noun also perhaps from German Spiel "play, game."

Source: Etymonline

Wiktionary hints at a possible relation with the Yiddish term shpil from which probably the alternative pronunciation originates:

• From the German Spiel ‎(“game, performance”), perhaps via Yiddish שפּיל ‎(shpil). Cognate with Old English spilian ‎(“to revel, play”).