In the isolated context of losing weight, it doesn't seem like exercise is essential. Or to state in less absolute terms, exercise has less influence than diet choices for purposes of losing weight and maintaining it.
In my case, I noticed that the pounds melted off when I stopped drinking sugary sodas and eating bread/snacks and switched to more vegetables with olive oil. In contrast , when I started exercising regularly, I actually gained more weight.
The thing that messes up the simplistic idea of "exercise burns calories so with 2nd law of thermodynamics you lose weight" ... is the body's overwhelming desire to maintain homeostasis. (https://www.google.com/search?q=homeostasis)
Homeostasis means that more exercise can cause more hunger and thus bigger meals which offsets the exercise so no weight is lost.
Likewise, deliberately eating less can cause the body to conserve energy and do less physical activity which re-balances the body to not lose any weight.
For many folks, a long-term way to adjust the homeostasis is to change the diet from high-carb to low-carb and switch out the 400-calorie Starbucks frappuccinos and sugary sodas to green tea and 0-calorie sparkling water. These diet alterations have more chances of success than maintaining the discipline of burning 500+ calories a week via exercise.
Exercise is essential for healthy heart and bone strength but there are counteracting factors when used as a tool to specifically lose weight.
> people pushing the idea that diet / food restriction alone is sufficient for weight loss
Diet/food restriction alone is strictly sufficient for weight loss: I'm pretty sure this is uncontroversially true, but it sounds like you're saying it isn't. Am I misunderstanding you?