Especially given the importance of teamwork. Competitive people tend towards tedious one-upmanship. It's something you want to screen out in the hiring process.
Programmers participate to ICPC in teams of three. According to my experiences, teamwork is a fairly important part of success since there is only one computer for each team.
I had to compete for my job, in which I get to spend a lot of time working relatively unconstrained on scientific problems of my choosing, as long as they're quite broadly mission-oriented. I like working on questions no one else is asking and avoiding the 'a large debate exists in the community as to whether X or Y' messes that Nature loves to publish.
I had to compete for my wife, who supports me and inspires me and etc. me, with few constraints (such as no others compete for our affections).
I hate games of all sorts (they have rules and the fun is to remember them and play by them?). But nonetheless I realize:
Elon Musk and Steve Jobs never spent time responding to competitors. VC funding is another example. Fundable companies are so rare that VCs never sit around debating whether to fund company A vs company B, it's two separate decisions.
It sounds like what you treasure most about your job is that you're released from competing. Finding a relationship is about fit, and a good relationship is about the investment you make. I've never seen competing to work in any phase of a relationship.