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.
It depends. If a method is well-known, but flawed (be it widely used in industry or not), then someone should point out its flaws. Not only is this the right way to progress science, but - depending on the applications - not doing so can be downright dangerous - let's say the method is used in clinical trials, or to hedge risk. A paper that dismantles an old one strikes me as the right approach, naming authors necessary as the method is associated with them. If the new method might replace the old one, the proper rebuttal would be publication - ideally in the same journal.
Where is the 'It depends?' The new paper should be civil, fact-based, and not personal, of course. It's about using a method properly, not about disparaging competing researchers. Additionally, if a new paper - like yours, say - is only an alternative method, not one that corrects errors, then the old paper should be part of the literature review, and the merits and dis-merits of the new paper highlighted, as well as situations where one or the other performs better, should this apply.
Papers like this certainly exist, and can be famous. Here is an example by David Kreps (Stanford GSB), which cast doubt not on a particular paper, but on an entire then-popular sub-discipline. It was certainly not written for any purpose other than academic curiosity, and a desire to question the current state of the art. While this is not exactly the same situation, I think it compares in spirit.
That someone tries to find flaws in your work does not mean they are trying to be "mean" or to belittle your work. It means you did something that seemed to be good and important enough for someone else to bother trying to investigate and improve it. Then the rational reaction is you should feel honored instead of sad.