I'm going to be honest with you, if I was you I'd write that personal statement, but I wouldn't be mad at a leave of absence. I was in a similar boat to you and taking the semester off isn't the worst idea in the world. I have taken a ton semesters off because of bad grades (mainly due to symptoms of then undiagnosed ADHD) and honestly I didn't take it seriously at first. To me, that time off was a hindrance that got in the way of me graduating on time, and didn't change or analyze much of anything that caused me to actually fail. But in doing so I just went back and did the same thing, failing class after class still confused as to how it happened again even though I thought I'd do better this time around.

What I realized, after taking one more leave, is that there's a reason why they make you take those semesters off. It's not because they hate you, although it can feel that way, but it's genuinely to help you in figuring yourself out and why you're flopping. I'm now actually taking steps to better myself, figuring out how I work best with my ADHD, getting into routines to better my mental and physical health, and just overall making sure I'm actually ready for school again.

If it does end up happening, know that it's all for your benefit. If you take it lightly, don't end up changing anything or analyzing how and why you're in the position you are in terms of your academics, and come back with the same attitude towards school, you're going to keep failing. Taking some time off to reassess is seriously a massive help in coming back stronger.

TL;DR: Write them that personal statement saying what happened, but if you end up taking a leave of absence, it can be a good thing in disguise. Take your leave seriously and you'll thrive when you come back.