I didn't mean you need to prioritize them over everyone else all the time, but I do think when you're starting a new relationship, you sometimes need to prioritize your relationship with that person over your other relationships just to let them know that you are serious about the relationship (if you are, indeed, serious. If you're not, just politely end it). For example, if you've been seeing your boyfriend or girlfriend for only three weeks, but you haven't been able to see them at all on that third week, you may have to actually tell your sister instead, "So sorry, but I have to miss your soccer ball game today. I haven't seen my boyfriend at all this week, I hope you understand." Your family is probably going to be a lot more understanding than the person you've been blowing off all week LOL. Though I do think your example of getting drinks after the soccer game is fine, too.
But that's what I mean by relationship strain to begin with. I'm not saying that the boyfriend or girlfriend is deliberately keeping you from seeing friends or family (that is abuse, and the relationship should be ended in that event), but I am saying that having to pick and choose who you spend time with can be very stressful. I'm not even saying that it creates severe or impactful strain, but it can create very small strain a lot of the time.
As for the puzzle piece thing, I think that's a nice saying, but it doesn't always align with real life. Having a relationship takes a lot of deliberate, hard work a lot of the time. By definition, you have to expand your time for that person (e.g., the puzzle board has to get bigger, and some of the pieces might have to get smaller) in order to fit them in, and yes, you do have less time for friends and family (unless you want the person to feel like they are also dating your friends and family LOL). I do think this is a mistake a lot of people make going into dating, too. They think the person they date should just already be perfect and fit into their life with no issues at all, and even if the person does nothing wrong and they may have been a great spouse, the relationship gets ended before it had a chance to flourish.
But to bring it back to the topic, I think @BoomerZ put it best. I think a lot of these stories written by younger people feel unrealistic and melodramatic a lot of the time because they have very little experience in love or maybe the last relationship they had happened in high school.