to answer the OP's title, they're not necessarily mutually exclusive.
There are different reasons for why people like the games they play. In my case, for example, I don't get the appeal of realistic graphics (maybe because.. 1 - too much detail would require a lower pacing, to avoid the details to become a distraction; 2 - I value the pacing when I'm reading manga or watching movies/anime/etc, and try to understand how do the writers manage to keep the viewer's attention; 3 - I've gotten used to having a low expectancy about my available time for reading written works and/or watching stuff, so I've gotten used to read spoilers or brief summaries or the beginnings and ends of books -- that says much about my expectations of having a good "reading experience".. and sadly, this low-fun-expectation of mine has reached videogames as well.), while gameplay mechanics (which offer lots of factors to consider, lots of virtues and flaws to exploit, lots of ways for attaining a goal, lots of fast decisions keeping me interested, entertained and mentally exercised) tend to appeal to me the most.
Ever since learning about how deep fighting games and shmups could get, gameplay-wise, I learned how trusting on some important information you know (for example, the timing and enemies' possible attack patterns in platformers such as Megaman. As for shmups, there are techniques such as hoarding bullets, not moving around needlessly, planning paths ahead, and other stuff regarding survival and scoring. For fighting games, there are factors to consider such as your execution accuracy level, reaction speed, foresight, perception, matchup knowledge/experience, spatial/temporal sense, ability to create situations in which you have higher rewards for lesser risks, how to try to make your opponent commit mistakes, or how to use game-specific mechanics besides fighting game fundamentals) and executing what you have planned based on such information can
be even more rewarding than playing without such knowledge, without such material from where to base decisions from.
Finally, about stories.. umm... maybe I would enjoy stories in games more, if I weren't exposed with enough stories which didn't seem to meet my "fun expectations". Reading LotR and not liking it, and playing Donkey Kong (arcade version) for score for a week and not liking it, does have some negative impact on my "fun expectation".