Is there such a thing as an unsafe router which is still unsafe after installing OpenWRT?

For example, privacy violating linksys or netgear, or devices with components running improper firmware with a 14 year old vulnerability?

The reason that I ask, although I don't want this to impact the quality of answers, is that I'm shopping for a new router that is secure and private but rather than paying commercial and industrial prices I would rather get a consumer router and overwrite it's software.