The larger the organization (private sector or not), the more likely it is to favor specialists, not least because smaller organizations simply cannot allow most of the people to be specialists (there aren't enough of them to do all the things).

So, if you want positions at large organizations, there will be more demand for specialists, and the reverse is true with small organizations. Whether you made a "mistake" or not, depends on whether you would rather work at small organizations, or large ones.

Note: yes, there are a minority of generalists at large organizations, and a few specialists at small ones, but the percentages are very different.