I do not believe that checking the assumptions of any model qualifies as p-hacking /fishing. In the first article, the author is talking about analysts who are repeatedly performing analyses on a data set and only reporting the best result. In other words, they are purposely portraying a biased picture of what is happening in the data.
Testing the assumptions of regression or any model is mandatory. What is not mandatory is to repeatedly re-sample from the data in order to ascertain the best possible outcome. Assuming researchers have a large enough sample to pull from, they will sometimes re-sample over and over again...perform hypothesis tests over and over again....until they achieve the result they want. Hence p-hacking. They're hacking the p-value via looking for the desired result and won't quit until they find it (fishing). So even if out of 100 hypothesis tests they only achieve 1 with a significant result, they'll report the p-value belonging to that particular test and omit all the others.
Does this make sense? When checking model assumptions, you're making sure that the model is appropriate for the data that you have. With p-hacking/fishing, you are endlessly searching the data/manipulating the study in order to achieve your desired outcome.
As for the purpose of multiple comparison, if you keep running a model through the mud endlessly trying to find a way to invalidate it (or validate it) then eventually you will find a way. This is fishing. If you want to validate a model, then you'll find a way. If you want to invalidate it, then you'll find a way. The key is to have an open mind and find out the truth - not just see what you want to see.