Is it possible to use the binary version of syncthing from a shared hosting , or it is always needed a vps with full root access?
It does not need root. Whether it works and is allowed on your shared hosting, only you and the shared hosting provider can figure out.
I run an instance on a shared hosting account (german only: https://uberspace.de/), they even have daemontools to run it as a service so no disadvantage for syncthing compared to a vps 
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