I wrote a piece on this exact point and how much you should actually worry about EQ:
Technical and cognitive skills are clearly "threshold factors" that cannot be ignored— they get you in the game and let you keep playing it. While they might not be sufficient, they sure are a necessary condition for success in most domains.
In order to stand out on your EQ skills you have to be first competitive on your technical/cognitive skills. It will be tough to compete just on your social skillset.
EQ is probably not a differentiator at lower levels and early stages of your career. Technical chops, cognitive skills and execution will probably help you stand out more. It's only when you move on to the managerial and executive ranks that EI/EQ starts becoming a differentiator, what Goleman calls a "discriminating competency".
Even at higher levels, EQ is not a given. Where it can probably make the most impact is in avoiding pitfalls once you get there, what researchers call “leadership derailment", rather than being an active mechanism in reaching there.
More here: "Misled and Oversold on Emotional Intelligence" https://www.leadingsapiens.com/ei-vs-iq-misperceptions/