Can Newton's Laws of Motion be considered the first principles of mechanics?

It depends on the situation. Within a certain scope, it absolutely is; but in a broader context, it is not.

You can understand it this way: imagine "first principles" as the fundamental rules at the bottom layer of a game, from which all gameplay and techniques are derived.

In the game of "classical mechanics," Newton's laws are the most fundamental rules.

Almost all motion phenomena we encounter in daily life, such as throwing a ball, an apple falling from a tree, how a car brakes, or how a rocket launches... all fall within the realm of "classical mechanics." Within this realm, Newton's three laws are the starting point and cornerstone for analyzing and solving all problems. We don't ask "why is force equal to mass times acceleration?"; we simply accept it and use it to derive everything else. Therefore, for classical mechanics, Newton's laws are indeed the "first principles" it deserves to be called.

However, the world of physics is much larger than what we observe daily.

When objects encounter two extreme situations, the rules of "classical mechanics" no longer apply, and we need to switch to a more fundamental set of rules:

When objects move extremely fast, approaching the speed of light: Newton's laws become inaccurate, and we must use Einstein's theory of relativity. Relativity has its own first principles, such as the "principle of the constancy of the speed of light."
When objects are extremely small, down to the atomic and electronic level: Newton's laws completely fail, and we must use quantum mechanics. Quantum mechanics also has its own set of first principles, which can even be described as profoundly counter-intuitive.

To summarize:

You can view Newton's laws as the foundation (first principles) of the grand edifice of "classical mechanics," where 99.9% of the mechanical problems in our daily lives reside. However, it is not the sole foundation for the entirety of physics (which includes all structures like relativity, quantum mechanics, etc.).

Therefore, if you aim to understand how the world around us operates, considering Newton's laws as the first principles of mechanics is entirely correct and highly effective.