No it doesn't. People say shit like, if he leaves CA and plays for a TX team he'll save like 12 percent! That's completely wrong. They pay tax in every state in which they they play, like you said. They also spend about two months in spring training. They can still maintain their residence wherever they want. There's no way he'll cut his rate in half because he'll spend about 90 days in TX.
But the main issue is that I'm 99 percent sure that this doesn't matter to the players. They want the biggest deal, not the biggest after-tax cash flow. He's going to earn over 300m in his career. He's not making decisions based on his state apportionment.
Sure, it doesnt work that way and while I agree with your general sentiment that are a number of over the top articles on this topic, the tweet also didnt say that It said he was going to no income tax state, which is true and in Degrom's case is adding a significant amount of after-tax cash. I think the most you can say about this tweet is basically the problem with all of Twitter - all complex issues are reduced to their most simplistic form due to character caps.
I dont think its the driving factor but guys clearly are advised on this and the habits of MLB players living in no income tax states makes me believe they are aware of what they are getting into (never mind the NFL practice of guys going to their no-income tax homes during bye weeks for apportionment purposes). Its probably not as big a deal with MLB players because there's less opportunity to apportion to the home team state (practice days for away weeks can be apportioned to the home state) where as MLB road trips are true out of state duty days, but the prevalence of all of these factors leads me to believe they know and consider these points in making their decisions. Are they going to turn down 5-10% more? Of course not, because they know that the impacts are never going to be that significant. But at the margins and as tie breakers? Sure, absolutely.
Edit - It may be that he was lucky in that he didnt go broader and doesnt understand, but sometimes its just as good to be lucky as good,.