
Really it comes down to if you have religious faith or not. You can absolutely partake in religious traditions from both and be part of both culturally, but when it comes to having the religious values they teach, there are significant aspects in both which contradict each other making it illogical to believe in both, mainly being that the concept of jesus does not fit standard jewish teachings.
i think a lot of conservative rabbis do a lot of outreach to patrilineal jews and want them to feel included. and others just don’t make a big deal out of it unless it has to do with like counting minyan or giving aliyahs. if a person is patrilineal and they want to be counted in a minyan or have an aliyah most rabbis would want them to convert to make it in accordance with halacha with but it would be a very different process than a person not raised jewish at all converting to judaism