How crucial is a storyline in a Waifu game? It depends. (a lot of my answers start like that)
I've played 'waifu' games with absolutely no storyline, but knock-out backstories for the characters you can romance. I loved them.
I've played 'waifu' games with expansive storylines that made me want to burn my computer and bleach my brain.
So let's try to be a bit more specific. (Yes, I've played enough of these games to get specific).
If you mean like a dating sim type, where you have to 'level up' certain character traits (these are typically 'strength', 'intelligence', and 'charisma', or some variation of them). These are the games where you typically have 3 choices, and each has a preference for the 'trait' you need to level up to romance them. They'll usually be in some kind of bar or club or something to that effect.
These are the kinds of games where you need to learn what a 'sweatdrop' means she needs on a date, and you need to remember awkward details. Some of it is realistic, like her birthday and what day she's free (in numbers, because they can't be bothered to code text recognition).
Story lines in these kinds of games are...well, kinda not the point. It's about 'doing' stuff, about 'levelling up' as a character. It's recognizing certain things and acting upon it.
Now, if you get more into...light novel type games, where you only click the text to progress the story? Then the storyline is the whole point. Skimp on the story, and you shoot yourself in the foot. But make no mistake, I've played games like this without a storyline... Please don't.
These games are more typified by the branching progression system. Where you can play through the game a dozen times and each time get a different ending, or end up with a different girl. In some games, I've ended up with more than one (and sometimes they play that off as kind of awkward afterwards).
Right, but then you have the 'mod' type games. Where you have a pre-existing game (The Elder Scrolls 5 Skyrim, for example) and you make a small 'update' of sorts, so people who already have/play the game, can add your mod and get extra content.
In these types of situations, it really depends. In some cases, adding a storyline is kind of pointless. Sometimes it's just about updating possibilities, like having kids (the old fashion way) or just being able to see the scene playing out. Sometimes it's adding a 'prostitution option' to dialogue.
In above cases, the game play is your storyline. Adding 'story' isn't always needed, sometimes it's just about adding options to choose from. So, it isn't about what's said, per se, but just that the NPCs (non-player characters) can do things they couldn't before, and those actions are the story added. But, in this case, you have the pre-existing game they can otherwise do, if the mod content is boring.
This, therefore, holds less important than a stand-alone game where the game itself is what draws in the players.
I just thought of a comparison that might make my point a bit clearer (yes, I ramble, yes I am aware of it). Let's look at Pokemon Red/Blue versus Pokemon Sun/Moon.
In both games your goal is to collect all Pokemon, and to beat the Pokemon League, right? Well, in Red/Blue, you'd be right. In Sun/Moon, you're just scratching the surface.
In Sun/Moon, you can also play with your Pokemon more directly. Give them treats, rub their heads, build islands for them, offer them more items...there's just so much more you can do.
These features don't add 'storyline', but it makes the game more immersive, and makes you want to keep playing, because you can 'do other stuff' other than just battle.
You see what I'm saying? If you apply that to any game, you give your player more reason to keep playing, because they have more functionality to keep them coming back. Without adding storyline.
Games don't 'need' a storyline to be great. Just look at Minecraft (the original, not the last updates where suddenly there's 'things'. Sometimes people just want stupid fun, where they can unplug and just do a thing to unwind.
Same goes for 'waifu' type games/content. Sometimes they want meaningful interactions (typically women, but not always/only), sometimes they just want fanservice (and yes, you know what I mean by that).
It depends. Who's your core market/audience? What do they want? Do you want to put in both 'story' and 'fanservice'?