I agree with your concern that sometimes "evidence based" is sometimes tossed out with less-than honorable intention. Just as on this website the use of "citation needed" varies between useful and snarky.
According to my memory and Wikipedia (proper citation needed), the term was in use in medicine before it was used in education. Ideally, it refers to policy and population-based medicine more than individual medicine, as patients have a nasty habit of getting a disease not fully studied.
In medicine, there is some push-back from those who think the role of intuition and common sense are now downplayed too much. See the following and all the follow-up:
 Smith, Gordon CS, and Jill P. Pell. "Parachute use to prevent death and major trauma related to gravitational challenge: systematic review of randomized controlled trials." BMJ 327.7429 (2003): 1459-1461.
In mathematics education (I am a math professor) there is similar tension, especially at the university level. There are lots of trials, some with good controls, of larger classes, like calculus or Newtonian mechanics 101. The small number of students who take more specialized classes, and the variations in student populations, classroom architecture, for example, mean that there is never going to be solid evidence from a trial that closely matches a lot of courses.
There is, however, a lot of informed writing on more advanced education that one might call a case studies. Given limited data, what can we say? For example, I have read things like the following:
 Braun, Benjamin, et al. "What Does Active Learning Mean For Mathematicians?." Notices of the AMS 64.2 (2017).
One can also look for items that cite an article, as there may be rebuttals.
In writing a teaching statement, or a curriculum re-design document, why not take the time to review some of the literature that is relevant? If you enjoy teaching, you should enjoy some of the education literature. Notice I said some. I would say the same about mathematics literature. Find a few sources you find informative and be sure to balance those out with common sense. Not everyone reading a teaching statement will be impressed by citations to the literature.