I work as an applied mathematician. In general, I would say that this is incorrect. Virtually all of the papers that I read I would contend have full proofs. That said, I can sympathize with the sentiment in a certain sense.
Just because a paper contains a proof doesn't mean that the proof is correct nor that it's comprehensible. Further, even if a paper went through peer review, it doesn't mean that it was actually reviewed. I'll break each of these down.
First, a proof is just an argument that an assertion is true or false. Just like with day to day language, there are good arguments and bad arguments. Theoretically, math contains an agreed upon set of notation and norms to make its language more precise, but most people don't abide by this. Simply, very, very few papers use the kind of notation that's read by proof assistant tools like Coq. This is the kind of metalanguage really required for that precise. Now, on top of the good and bad argument contention, I would also argue that there's a kind of culture and arrogance associated with how the community writes proofs. Some years back, I had a coauthor screaming at me in his office because I insisted that every line in a sequence of algebraic reductions remain in the paper with labels. His contention was that it was condescending to him and the readers to have these reductions. My contention was that I, as the author of the proof, couldn't figure out what was going on without them and if I couldn't figure it out with all those details that I sincerely doubt the readers could either. Around the office, there was a fair amount of support for my coauthor and removing details of the proof. This gives an idea of the kind of people in the community. For the record, the reductions remained in the submitted and published paper. Now, say we removed all of these steps. Did we still have a full proof? Technically yes, but I would call it hateful because it would require a hateful amount of work by the readers to figure out what was going on.
Second, peer review is tricky and often incredibly biased. Every math journal I've seen asks the authors to submit to a single blind review meaning that the authors don't know their reviewers, but the reviewers know the authors. If you are well known and well liked in the field, you will receive the benefit of the doubt if not a complete pass on submitted work. I've seen editors call and scream at reviewers who gave "famous" people bad reviews. I feel like I was blacklisted from one community because I rejected a paper from another "famous" person who tried to republish one of their previous papers almost verbatim. In short, there's a huge amount of politics that goes into the review process. Further, depending on the journal, sometimes papers are not reviewed at all. Sometimes, when you see the words "communicated by so-and-so" it means that so-and-so vouched for the authenticity of the paper, so it was immediately accepted for publication without review. Again, it varies and this is not universal, but it exists.
What can be done? I think two things could be done immediately and would have a positive affect. First, all reviews should be double blind, including to the editor. Meaning, there is absolutely no good reason why the editor or the reviewers should know who wrote the paper. Yes, they may be able to figure it out, but beyond that names should be stripped prior to review and readded only at publication. Second, arbitrary page limits should be removed. No, we don't need rambling papers. If a paper is rambling it should be rejected as rambling. However, it removes one incentive to produce difficult to follow proofs since now all details can remain. Virtually all papers are published electronically. Page counts don't matter.
In the long run, I support the continued development of proof assistant tools like Coq and Isabelle. At the moment, I find them incredibly difficult to use and I have no idea how I'd use them to prove anything in my field, but someday that may change. At that point, we can remove much of the imprecision that reviewers introduce into the process.
Can someone in the field comment?