Part 1, question 2, article 3 from Aquinas, Thomas: Summa Theologiae enumerates five ways to prove the existence of God:
1) God as the first mover (primum movens)
2) God as first cause (causa efficiens prima)
3) God as necessary by himself (necessarium per se)
4) God as cause of being, and goodness (causa esse, bonitatis, perfectionis)
5) God as the designer of the ends (causa finalis).
The first way is taken from Aristotle’s Physics, book VII and VIII. The second way is taken from Aristotle’s Metaphysics, book II. The third way is similar to a reasoning from Aristotle’s Metaphysics, book XII. But Thomas changes the argument supporting the claim that not everything passes away.
I agree, inductive reasoning and starting the chain of reasoning with a first cause is common to all five ways, not only to the first three.
Thomas did not aim to generalize all "proofs" to the abstract principle of the impossibility of an infinite regress. Instead, his aim was always to recall explicitly as many arguments as possible from Aristotle. Because Thomas wants to show that a huge part of Aristotle’s philosophy (reason) justifies the argumentation from theological statements (faith).
In addition, if people are not used to the abstract level of principles, possibly one can convince them by appealing to specific and concrete examples of a principle in question.