I myself am involved in the writing side of a game in development, so of course writing is important to me, and the quality should be high.
whether I find it essential for it to be as correct as possible, really depends on the nature and overall quality of the game itself. There are games that I play purely for the visuals or kinks "in play".. and I might even skip dialogues once I've got the gist of the setting or scene.
other times I find the writing to be incorrect but broadly accessible and not too terrible; clearly the product of a non native speaker (such as myself)..and I appreciate the effort.
Then there are games that have clearly been written in another language and roughly thrown into google translate, or a worse iteration of the same.. without the least attempt at proofreading or verifying if the text makes sense. that's a big no no for me... especially so if they've been around long enough and have a big enough fanbase for the dev to have had ample opportunity to rope in a volunteer proofreader.
The main thing for me is a coherent plot and believable characters, possibly with some originality to it... if that's dealt with with a modicum of competency, typos and nonsensical English, whilst grating, aren't always dealbreakers..
But if the plot is lacking, nonexistent or just too basic, no amount of Shakespearean flourishes can save it from being skipped through to get to the juicy bits... provided those are of an adequate quality... if not, the whole thing goes to the bin.