Hey Jeff, welcome. Mildew/algae are one of the unfortunates of exterior paint. Zinsser guarantees mildew free for five years, so for what that's worth?? Give it a shot, it can't hurt. As to the peeling, be sure that all joints in the wood are properly caulked. Technically, all wood joins should be caulked, whether they're gapped or not. What's tight today may not be tomorrow. Moisture getting into/behind the structure is the single biggest cause of paint failure. Caulking occurs after priming/first coat of stain to prevent absorption of caulk vehicle and premature failure. As to covering painted surfaces with a stain, you can. Just be sure to scrape, sand, and properly prime all surfaces. You don't need to prime if you go the stain route. Also, after you prep be sure you give ample time, couple of good drying days, for any exposed wood to dry out before recoating. A lot of people prime immediately, sealing residual moisture in which only guarantees repeat failure. If you're certain the wood is dry you're okay. Good luck.
Joe
PS Why would putting a solid stain over a solid painted surface look stupid? There's no need to strip any paint. With the proper prep, the stain will bond to the paint.
Joe
PS Why would putting a solid stain over a solid painted surface look stupid? There's no need to strip any paint. With the proper prep, the stain will bond to the paint.