Papers written with the sole purpose of finding flaws in someone else's paper
Whether papers like this will be published depends on the journal and the editor. It usually is preferable to show how a problem can be solved correctly. However, it might be well justified to solely point out weaknesses in existing work.

Keep in mind that very few things are actually "bullet-proof". The more you know about a method (new or old) the more this will become obvious. Most methods have their limitations. Newton's laws are still "pretty good" although Einstein and others have shown their limitations.

If you have confidence in your new method, you will probably highlight the advantages it has over existing methods in your paper. However, you should also be open about the limitations of your own contribution. Be clear about what your method can and cannot do and where and when it is applicable. In being honest about your own work you are making a true contribution to science. However, if you’re starting to have severe second thoughts about your method, it might be better to double check what you're doing. Talk to other people in your field and openly discuss your concerns before publishing.

As long as you don't over-egg your own contribution you will be close to bullet-proof. It won't be bad for your reputation or career if somebody were to find a weakness in your work and improve your method or the knowledge of the field. That's part of how science works. You can only make contributions to the best of your current knowledge and move on when you have new information. The only thing negative for your career would be to ignore new evidence and trying to hide past mistakes.