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The problem with trying to fix speedy deletion is that most proposals to "fix" speedy deletion suck. As a result, people generally take such proposals with a grain of salt. Some suggestions to test proposal ideas based on what I've observed from previous proposals:
Minimal extra bureaucracy. Many proposals try to "fix" speedy deletion by throwing in a ton of extra checks and balances with the end result of a system that's more complicated than AFD.
Focus on all the problems. The problem is not just careless deletions. Many careless deletions are caused by careless tagging, which is caused by the massive amount of bad pages created 24/7 and a lack of people to adequately patrol them.
Don't just shift the work. A lot of proposals try to just push the more subjective deletions onto other deletion processes. About as many articles are speedy deleted in an hour or 2 as are AFD'd in an entire day. Dumping the load on another system is not a solution, its just creating a different problem.
Non-admins cannot see deleted content. This is basically straight from the foundation's legal counsel[1] and is pretty much non-negotiable.
Look at the problem in perspective. 5 bad speedy deletions (a completely made-up number) per day sounds like a lot, but bad speedy deletions probably make up about <1% of all article speedy deletions and maybe <0.1% of all speedy deletions. A complete overhaul to fix the corner cases is overkill. Don't try to kill a fly with a nuclear bomb.
Look at implementation costs. A new policy that requires dozens of people to radically change what they're doing, and requires massive infrastructure changes is much more likely to fail simply because it would be hard to implement, regardless of how good an idea it really is.
The key word is "speedy." A speedy deletion system that takes more than a day to delete an obviously crappy and unsalvageable page is going to be unacceptably slow.[2]
Poor fit with other policies. Many proposals suggest implementing things that are directly contrary to policies other than CSD. The odds of successfully making a significant change to one policy are low enough, trying to change several at once is near impossible (and is probably also overthinking the problem and failing test number 5)
Not all speedy deletions are equal. Deletions done for BLP reasons should not be overturned without a clear consensus to do so.[3] Deletions done for copyvio reasons have similar restrictions. Any proposal that attempts to delay speedy deletions or make overturning easier needs to make exceptions for, at minimum, BLP vios and copyvios.[4] Valid G4 speedy deletions are also similar, in that since there was a discussion to delete, they should not be restored without discussion.
Don't propose vaporware. A proposal suggesting changes to the technical side of things should make sure that any technical aspects not yet available are feasible and able to be implemented quickly. If you can't do it yourself, make sure you find someone who can and will. This is also related to test number 6; the developers aren't going to rewrite the parser and disable caching just so a template the proposal needs will work correctly.
Any proposal for a significant change that doesn't take these issues into consideration is, while not guaranteed to, very likely to fail. A replacement system that is arguably worse in some areas than the current system while better in others is basically replacing a mediocre system with another mediocre system - something that's not worth the effort.