This is kind of an interesting question. Three observations spring to mind.
First, you're really not going to short-circuit the need to present the basic inverses method of solving (the addition and multiplication properties). Your factoring examples have skipped presenting those steps, but they're still in there, e.g.:
$$2(x-3) = 0 \implies x - 3 = 0 \implies x = 3$$
You definitely need to explain and justify that middle step (in which you'll add 3 to both sides of the equation). I mean: sometimes test-preparation materials can cheat on this, maybe just teach by rote that if you see $(x-3)$ as a factor you'll get a solution of $3$, but that would be invalid mathematics and students would suffer later on with that kind of "faith based math".
So you still need to deliver the basic inverses technique even to finish off your examples of solving by factoring. If you expect to also teach the method of factoring and the zero-product property, then at that point it seems like an unnecessary detour just to solve a linear equation.
Second, you can be sort of tricked by lots of "nice" examples that are being given to make life easy for the beginning students. Sure, many starting examples will have the constant term divisible by the linear coefficient (i.e.: factorable in integers, which is another unstated assumption). But what about any other case? E.g.: $2x - 3 = 0$? Again, you immediately need both the addition and multiplication principles in order to finish that off. What about general numerical problems: arbitrary fractions for coefficients, arbitrary decimals, etc.?
Note that many or most algebra books quickly exercise students on such general linear equations, ones that cannot be factored in integers. At this point you have a fairly nice general technique for solving linear equations of all sorts. For example, see OpenStax Elementary Algebra, Section 2.5: "Solve Equations with Fractions or Decimals", which comes immediately after the general strategy for solving linear equations by inverses.
Third, many books and curricula also treat general linear inequalities at about the same time. That's pretty close to the same process, with one added trick (flip inequality direction if one multiplies by a negative number). It's even less clear what kind of trick you'd apply to jump over that "missing" step in your examples to handle this with an always-factoring approach. Again, see OpenStax Elementary Algebra, Section 2.7, for these applications.
(Note also that this curriculum then follows with graphing lines and solving linear equations before higher-degree objects are handled; this provides a spiral-type path where you get to revisit the ideas of solving equations, inequalities, and graphing, in progressively more advanced contexts -- which is often needed by such basic students.)
In short, the general process for solving linear equations and inequalities can pretty quickly be presented, and in fact must be presented, even if you wanted to focus on factoring all the time (which therefore presents an unnecessary delay). So the student has a fairly nice package of tools to handle linear stuff, possibly numerically with calculator technology, even if it's not factorable in integers.
In fact, for some students they may not progress any further at all in their math pathway. Consider in this case OpenStax Prealgebra: that work manages to cover solving linear equations, but never gets to any higher-degree work. For some students that will be the end of the line, and time spent on factoring will be an unhelpful delay and distraction. (A key point of debate for basic math skills at my institution has in fact been administrators arguing that non-STEM students don't need to learn factoring, for example.)