I've thought about this as well so you're not alone. I am in no way qualified to answer but I would think the needle gauge they use would be too small, you'd clot before you lost a significant amount of blood. Someone who knows what they're talking about feel free to correct me.
Something I have thought about, if one had unlimited time and privacy and a lot of will, would be a slower attempt at exsanguination. Like maybe you can't hit a vein or artery and bleed out in one go, but you can still inflict a fairly major wound and bleed a lot, and keep doing that until you reach critical blood loss. Maybe in combination with drowning. This seems pretty implausible, I imagine SI would kick in and you'd end up calling for help. But maybe.