The way I understand it if you want transparency using just VDP1 in hardware, only items displayed by VDP2 will be seen through it. VDP2 can do hardware transparency, but I'm not sure it's variable. It's usually just used for background layers but, if I understand it correctly, the fire in burning rangers is rendered by VDP1 and then transferred to VDP2 and displayed on screen as transparent. That certainly explains why the fire has different than normal depth sorting issues. Also, there's apparently a performance cost to this method that makes it not ideal for every game.
I'm not sure how D-Xhird is achieving the transparent skirt, because legs are visible but not the background. It could be the same as Burning Rangers or it could just be rendered in software. Also, that game pulls off some impressive effects but overall, it's not very fun and the character design and animation really hurts it's presentation. When I first saw it on youtube I thought it looked really impressive but I think the compression really helped gloss over a lot of the games shortcomings. This was back when 360p was the normal quality of youtube videos.
At the end of the day transparency on Saturn isn't an impossibility, it's just not convenient, versatile or cheap. At least not when you compare it to PSX or N64.