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

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”).

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.

JOSH's answer resolves the central question posed by MrHen. With regard to similar words adopted from German into English without inclusion of an h in the anglicized spelling, perhaps the closest match to spiel is spritz. Here is the entry for that word in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

spritz \'sprits, 'shprits\ vb {G fr. spritzen} (1902) : spray ~ vi : to disperse or apply a spray

According to Leo Rosten, The Joys of Yinglish (1989) there is a Yiddish-English form of spritz, and he spells it with an h (shpritz):

shpritz (verb and noun) shpritzer (noun) Yinglish. From German/Yiddish: spritzen: "to sprinkle," "to spray," "to squirt."

Rosten doesn't list a Yiddish/English equivalent form of spiel. However, The American Heritage Dictionary of the English Language, fifth edition (2011) cites the Yiddish shpil (derived from Middle High German spil) as a possible direct source of spiel in English:

spiel (spēl, shpēl) Informal n. A lengthy or extravagant speech or argument usually intended to persuade. intr. & tr.v. spieled, spieling, spiels To talk or say something at length or extravagantly. {German, play, or Yiddish, shpil, both [from] Middle High German spil [from] Old High German.}

The Eleventh Collegiate dates spiel in English to 1870 and gives only one pronunciation for it: \'spēl\ . This surprises me because I have heard \'shpēl\ frequently enough to think of it as a common alternative pronunciation. Likewise the Eleventh Collegiate says that the second syllable of the curling term bonspiel is pronounced \'spēl\ (not \'shpēl\ )—although it certainly was pronounced with a sh sound in Calgary, Alberta, in the early 1970s, when I lived there. In any case, Merriam-Webster thinks that bonspiel, may be derived from Dutch bond (league) + spel (game); this word's first known occurrence in English is ca. 1770, so it has been in the language much longer than the standalone German-derived spiel.

Any discussion of English spiel should disregard Yiddish, which is irrelevant to its etymology. The word comes solely from German.

For details, see pages 563-570 of David L. Gold's Studies in Etymology and Etiology (With Emphasis on Germanic, Jewish, Romance, and Slavic Languages).