Although I find the tone of your post a bit too "ranty", you address some important points. My first feeling is Academia StackExchange would be a better fit for your question, as it is a more philosophical question that affects all fields of science, not only bioinformatics.
Your post addresses several issues about modern science:
- how to fund research (in general, and more specifically, research publication)
- how to evaluate research
- how to evaluate researchers competence
I will address only 1), and briefly and in a over-simplifying way at that. As the saying goes, there is no free lunch, and publication process has to be paid somehow. There are two competing models right now (with possibly a third in the making): 1. publication is (mostly) free, access to papers is paid - this is the big publishers (Elsevier, Springer, Wiley, and so on) model; 2. publication is paid, access is free - PLoS model; and 3. publication and access are free, with a simplified publishing process - arXiv / bioRxiv models.
As science is somewhat conservative, preferred models are 1) and 2), as 3) is not peer-reviewed, nor has an impact factor, etc - probably the fact it hasn't been monetized also counts.
To get to the point, I am wondering if there are journals that accepts scientific manuscripts without a mandatory payment.
Yes, several, those following the monetary model 1) above. You will often have to read a lot of "instructions to authors" and related pages to discover if and how much are the costs to publishing at a particular journal.
Or you may go the bioRxiv (also tweet / post about your pre-prints), if you are lucky, it will pick up peoples interests and several (mostly young) researchers will consider this as a good publication.