Were it this single verse, I'd be inclined to agree with your reasoning. Nevertheless, the parallelism between this verse (particularly since Christ uses the universal "anything"), and the various restatements of the principle in John 14 & 16, (lacking the possibly mitigating context) is inescapable.
Again, I think you are slipping in a third premise in that you know by experience this principle is not universal, despite explicit language, while appropriating as universals promises with far less supporting language simply because there is no specific negation of their universality.
And while restricting exorcism to clergy trained in same should be understood to be wise practice, it's not the same as making the claim only the clergy are capable. Much of Catholic practice is misunderstood in that it confuses prudence with injunction, by both Protestants and Catholics. Note that Catholic children (at least in the past) were trained to perform baptism on younger siblings should death be immanent and no clergy were available.
Is this not at least tacit recognition of the Christians' authority with a preference for the clergy in it's practical exercise?
Like I said before, not everyone is called to a deliverance ministry, any more than everyone is called to be an EMT--but knowing the basics of spiritual warfare is for everyone, just like knowing CPR.
A certain amount of training is certainly useful--I don't knock it and am grateful for what I have received--but the Spiritual gifts of discernment, words of knowledge and wisdom, tongues (strangely enough), faith, healing, and prophecy are far more important than any "book learnin'." When God calls someone to this ministry, He equips them. Surprisingly few of those I know in this ministry are full-time, paid professional priests, pastors, and ministers. My mentor, for example, is on the board of elders at my church and has a masters in Christian counseling, but he is an architect by training and vocation.
So again, if we're just talking about preferences rather than trying to dogmatically control who the Spirit calls to do what, I don't think we're too far apart in our stances--though my real preference would be that every Christian be trained in spiritual warfare and deliverance, that they would walk in faith and in the Spirit, and that they would be unashamed and unafraid to reclaim ground from the Enemy. But I'm also a realist (strangely enough) . . .
However, if you want to claim that all Christians have not been given authority over the demonic despite the Scriptural citations in my previous post, you need to build a Scriptural case against that position. It is not enough to draw a parallel to an unrelated passage (especially since I have already answered said parallel). At best, that only proves that authority over spirits may not be given to all despite the clear language otherwise, not that it is not.