First postdoc as "the big filter": myth or fact?
Like many other things in academia, this depends.
Having an unproductive postdoc is a point that will definitely come up with any future hiring committee. I’ve sat on a few myself (CS departments) and productivity is considered at every period of your academic life. Of course, if a candidate had a stellar PhD it will make up for it to an extent, but it’s entirely possible that such a candidate will be passed over in favor of a candidate with a less impressive PhD but a more consistent performance overall.
Such “performance gaps” always raise questions - does this indicate an issue with working independently? Working in a group?
Another point to consider is reference letters. Your postdoc host is not likely to write you a glowing letter if you are unproductive, which will hurt your chances of landing a position later on.
That being said, research does not always result in publications. If you made progress on an important problem and have at least a preprint/workshop publication, this may be enough. Hence, your focus should be on productivity rather than on the end product.
A postdoc is also an excellent time to show your commitment to the research community. Organizing workshops, taking on higher-ranking roles in program committees, setting up tutorials, writing a useful blog... All excellent ways of putting yourself out there and making yourself more marketable. Of course - research should absolutely come first, as it is the main evaluation criterion.
Remember - you successfully completed your PhD, you can (at least according to empirical evidence) succeed in this as well! Good luck!