The Importance of Priesthood

Both Catholicism (or Orthodoxy) and Mormonism believe that having a valid priesthood is vitally important. Both denominations also recognize the importance of lineage, and that the priesthood must necessarily be conferred by direct decent.

Europe, traditionally, viewed descent and lineage as vitally important. So too did the Jewish world, as the importance of Jesus' direct descent from David was considered critical in acknowledging His validity as Messiah and King of Israel.

The modern world has largely attacked this notion, as the essentially materialistic and communist notions underlying the present worldview are anti-hierarchical, egalitarian, and opposed to any notion of nobility or aristocracy - therefore acknowledging the importance of lineage is mostly viewed as forbidden.

Catholicism recognizes an individual priest's authority as deriving in a direct lineage from Christ, through Peter.  Mormonism recognizes an individual priesthood holder's authority as coming directly from Christ through the Apostles Peter, James, and John conferred directly on Joseph Smith during the restoration.

There are many differences from this point on, but the unity in emphasis is impressive. Having a valid, royal, prophetic priesthood is important.

My impression from this is that, for a large organization to be properly stable long-term and oriented towards God, it must have valid priesthood.

So, for example, while individual Christians without access to a priest may be real Christians - and they may even meet together and praise God in a real, vital way - for the organization itself to be ensured longevity and validity, it must have a valid priesthood.

Or at least, without one, it's missing something wonderful.

So then, my sense is that if an individual desires to join a church and submit to that church's authority in religious matters, they need make sure the church has valid priesthood holders who claim a direct, lineal descent from Christ.

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For those living in the Western world then, their best options are likely Catholic or Mormon. Unfortunately, the post-consicular Catholic church is in a period of self-contradiction, confusion, and disunity. In its long history though, the Catholic church has had periods of trouble, with some very bad Popes, so focusing on tradition (perhaps more akin to how Orthodoxy ensures validity) might be necessary. 

A local priest I spoke to said that, even with all those problems, the core of the mass and validity of its sacrament are essentially the timeless essence of that faith. So largely ignoring the present "winds" and instead focusing on Mass (the traditional Latin ones, especially high-masses, are quite beautiful) and traditional teachings are probably the best option for someone who wants to remain Catholic.