I am coming from a web application developer career, so no sysadmin skills. But I also like security topics.
I was wondering if after getting a certification such as GIAC, GPEN or EC Council CEH, or any other pentesting certification would make me legitimate or at least job-ready for the skills they are supposed to teach?
No, I don't think that the certification itself will make you legitimate and/or job-ready. However, you can learn a lot while preparing yourself for those exams. You haven't stated what you try to achieve, but by simply attaching a certification or two to your application, will not make you a great and experienced infosec guy, and every big/serious IT company knows this.
Don't get me wrong: It is definitely helpful in getting across your experience, but in the end there will be job interviews and you'll have to prove that you can stick up to the claims made in your CV. So, while it might be a good door-opener and a good foundation to start the discussion, you still will be asked about previous work experience, etc. Nobody will take it you for real, simply based on some certification attached to your application.
Unfortunately getting a decent system administrator is nothing you can learn and be done with it. It is a process of learning from mistakes you (and others) did. If you want to switch into this area (system administration, security) you should start small-ish, with some test setups / environments, read about it and shouldn't be too picky about any job.
I would argue, that Working in the infosec area is even more challenging. If you are seriously interested in it, just start to look into it, work through some material (there is a ton of it available) and try to apply it to some software you like. While doing this, you might get some attention from the company you want to work with, and applying for a job with them, might be become easier in the future.
Certificates do not make anyone legitimate or job-ready.
The skills that those certifications are meant to asses are what makes someone legitimate and job-ready.
Certificates are meant to indicate that you have experience in those areas.
