All consonants in a consonant cluster will be slender or broad together. The t is slender, so the s must be too.
Sráid has a broad s, but speakers who haven't mastered Irish phonology tend to say it like a slender s here, presumably because "sr.." is not a possible syllable onset in English, but "shr..." is.
I've noticed in recent years a tendency for English speakers (including many younger people) to use the sh pronunciation in words where a consonant comes second. Stop becomes shtop, Stink becomes shtink and so forth. German got there years ago, with Strasse pronounced [shtrassa] and Stein pronounced [shtine]. I suspect it is a natural progression in language. The only thing that will stop it happening in Irish is the broad/slender distinction, but as others have noted, even this is not guaranteed..
Those "rules" for pronunciation that you've been reading? They aren't rules, they're guidelines. In a case like this, the sibilance of the slender "s" is likely to mask the slenderness of the "t" anyway, but you can listen to other examples of "stéig" on teanglann.ie (http://www.teanglann.ie/en/fuaim/st%c3%a9ig) as well as other words starting with "téi" and "st" for comparison.