Yes, it seems that you're right. I've finally found a possible explanation on this forum. I'll translate and report here the answer of user maksymus (my additions in red):
From sъ-tlъkati («с-толкати», compare with Russian столкнуть «to push off»). The labialization of л to в is well recognizable (compare, e.g., Russian волк 'wolf' and колбаса 'sausage' to Ukrainain вовк, ковбаса). The transformation into [š] was, perhaps, analogous to шкаралупа (=eggshell, Russian скорлупа), кошлатий (=shaggy, Russian косматый), шпиця (=spoke, rung, Russian спица); possibly under Polish influence (compare: штурхати '(Ukr.) to push'— szturchać '(Pol.),'to prod, to nudge' ).
BTW according to the Polish dictionary by Boryś szturchać comes probably from Middle High-German 'stürn' 'to dig around, to pick, to poke' so it's not directly related to штовхати.
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I wouldn't seriously consider the derivation from тълкати: everything may have happened in the real life, but the change k>x is too aberrant for Slavic (I can't recall any example) to be acceptable for a good etymology. A contamination of two roots (like *стовкнути × szturchać) looks more promising.