The Olympus 35 EC does not have a mechanism to prevent activation of the shutter release if there is no power -- that was the major improvement of the EC-2. On the original EC, the only way to know if the shutter is actually working is to be sure the yellow lamp on the top illuminates when the release button is pressed part way down. Otherwise you can blithely wind and snap through an entire roll, but the shutter may never open.
If the yellow lamp never illuminates, it's likely power failure -- if not bad batteries, check the battery contacts inside the battery chamber, as well as the battery cover itself. Type 640 batteries were notorious for leaking and corroding the contacts and destroying the wires going to the camera circuit, and flooding the battery cover itself with corrosion.
If you have a voltmeter, check resistance between the outer metal of the battery cover and the chrome battery contact itself. If there is continuity, you can carefully pry up the black plastic plate glued to the metal part of the battery cover, clean any corrosion and put some insulating tape where the battery contact plate is exposed on the underside of the plastic, and glue the plastic plate back in place. If you suspect corrosion has invaded the battery contacts themselves (almost guaranteed), you can carefully pry up the battery chamber (once the bottom cover has been removed) and access the wires leading to the circuit board. The wires (one red, one blue) should lead to a solder joint within an insulating sleeve -- from there I'll leave it up to you how to will "install" replacement contacts, since you're likely going to be using something other than original 640's.