No. People accept pain or forgo pleasure for other reasons. You can argue that those higher reasons are about a different kind of pleasure, but you would have a hard time proving it. Even if those things are about some kind of pleasure, it is often not the kind of pleasure that is a pure, reductive decision to release dopamine and feel differently in the moment. In the simplest case it may be about future possible pleasure that most likely never occurs. Increasing the odds of feeling better later is logical, but not necessarily in a way pleasant in the moment.
Even if you include all possible future pleasures, experienced or merely planned, that produces so many dimensions of choice that you can hardly consider the person planning and arranging them to be forced onto a given trajectory. So if you want an argument about free will out of this, you can't get one unless you imagine people have a drive toward optimization or perfect reckoning that we all clearly lack. We choose sub-optimal courses at random all the time. The fact that there is no purely logical reason to act is unrelated to whether that action is freely chosen.
Besides that, why put logic and pleasure at odds. Logic itself is just another form of pleasure. The feeling of 'consilience' or settledness that arises from agreement with yourself is not in principle different from all other actions we take to avoid fear or distress and make ourselves comfortable. It is just far more subtle, and it is easier to learn to control it.
But we do not even consistently choose to pursue the pleasure of logic when other opportunities for pleasure are balanced. People balance logic against boredom, various kinds of stress and other conflicting emotions all the time. They go out of their way to do illogical things as assertions of their freedom, as sources of different perspective, or as tests of their flexibility, or their courage. They chose change for its own sake as a way of not being controlled by other people, or by situations. How much to value each of these alternatives in any given moment is not predetermined or directly dictated by their intensity.
These concepts do not cohere. Logic, freedom and goals may feed into one another in a positive way, but they are not linked up in some deterministic fashion, and the evidence is obvious in everyday life.