The vast majority of irreproducible papers aren't detectible as irreproducible at time of publication. They look fine, and many actually are fine. They just don't reproduce. That's an expected outcome in science. The system will self-correct over time.
IMO, the main actual problem with peer review is that non-practitioners put too much faith in it. Nobody in science actually takes a paper on faith because it's been published, and you shouldn't either. Peer review is little more than a lightweight safeguard against complete nonsense being published. It barely works for that. Just because you found a paper doesn't mean you should believe it. You have to understand it.
A secondary actual problem is that it's impossible to reproduce a lot of papers, or they're methodologically broken from the start (e.g. RCTs that are not pre-registered, or observational studies without control groups). These are problems we could actually solve. For example, just requiring that any paper publish the raw study data would help to self-control the system. There are high-profile researchers out there, right now, who do little more than statistically mine the same secret data set -- these people are likely publishing crap, but we have no way to prove it, because the data is secret.