Fisher's test is as bad as everyone says it is from a Neyman-Pearson point of view and if you do what your question implies---after a significant ANOVA test each individual difference. You can see this in many published papers. But, testing all the differences after an ANOVA, or any of them, is neither necessary nor recommended. And, Fisher's test wasn't crafted under a Neyman-Pearson theory of statistical inference.
It is important to keep in mind that, when Fisher proposed the LSD, he didn't really consider multiple testing an important problem because he didn't consider the significance cutoff a hard and fast rule for deciding whether results were important or not. One could construct an LSD as an easy way to peruse the data for where there might be significant results but not the arbiter of what was meaningful. Remember, it was Fisher who said that you should just run more subjects if p > 0.05.
And why would you think that testing everything is a good idea? Consider why you run an ANOVA in the first place. You were probably taught that it's because running multiple t-tests is problematic, as you intimate in your question. Then why are you running them, or their equivalent afterward? I know it happens but I have yet to ever need to run a test after an ANOVA. An ANOVA tells you that your pattern of data is not a set of equal values, that there may be some meaning in there. Many people get hung up on the caution that the test does not tell you where the meaningful bits are but they forget that the data, and theories, tell you that.