Only the "verify" part matters
When you have a putative answer to some problem, the only thing that matters formally is that you have a verification that the answer is an answer.
All the algebraic work you did when trying to solve the problem? Completely irrelevant — its only value is that it led you to something that you might then try to verify. (although, sometimes the work done can be reused in the verification step)
Since "guess and verify" includes "verify", it's a perfectly valid technique.
Now, we should understand why one might be uncomfortable with "guess and verify" as a technique.
People often focus solely on the "How do I come up with a putative answer?" part of problem solving, often to the point of forgetting the need for verification, or not even knowing it's something that needs to be done!
This is further exacerbated by the fact there are often tricks to verify a putative answer by reusing the work has done in coming up with the putative answer.
For example, when solving the equation $5x + 3 = 13$, one might do the steps
- $5x+3 = 13$
- $5x = 10$
Then, to verify $x=2$ is a solution, one can then observe that every step in this derivation is reversible; simply writing them in the reverse order amounts to a verification!
(that said, as a practical matter it's better to do verification by plugging $x=2$ into the original equation; this gives you a very good chance of catching any arithmetic mistakes)
If most of the problems have solved are like this, it makes verification almost look like an afterthought or a technicality, leading people to underestimate its importance, or even completely overlook it as a step!
Furthermore, one spends a lot of time learning how to do these sorts of algebraic manipulations to arrive at a specific result; this is only one particular strategy for coming up with a putative answer, but this focus often leaves people with the impression it is the only valid strategy.
The way math is taught might even reinforce this misconception; e.g. in a test of how well you can do algebraic manipulations, a guess-and-check solution would get marked down because you didn't demonstrate your skill at algebraic manipulation... but the student might misunderstand and think that they got marked down because guess-and-check is an invalid problem solving technique!