I understand the sentiment, and I'm the first to challenge the use of social classes as rather meaningless in everyday conversation. People tend to be far too loose with their concepts and 'middle class' doesn't have much explanatory power. But with this limitation in mind, it can be useful shorthand, especially when it is understood that there are relatively few of the proletariat in the global north – some might say none.
In many cases, the better off workers are labour aristocrats(LA) or petite bourgeois (PB) (which is true of a significant proportion in the global north). Those who identify as middle class tend to be LA/PB. Their interests are more broadly aligned with the haute bourgeoisie, too. Not so much as their interests are aligned with other LA/PB but, often, more than they are aligned with the world-proletariat. Enough to make them fear losing their privileges and to fight for capital.
Almost anyone with a pension, for example, is in a real bind. They most likely invest in fossil fuels (and possibly arms, tobacco, alcohol, patents, pharmaceuticals, etc). As much as they may fear climate change, they also fear having no pension after contributing for a lifetime. Faced with this contradiction, their motives are only fundamentally opposed to those of even fossil capital some of the time.
I agree that in many cases, the purpose of using the 'middle class' as a concept is to conflate several materially distinct classes. But the conflation of better-paid workers with LA/PB is accurate more often than not, in the global north. When middle class does equate to LA/PB, is functionally to exploit the proletariat, even if they are not really in control of the way that they do so.