Why would the state be deciding who is qualified to be an officiant?
For civil purposes, think of the officiant as an additional witness or co-signer of the license - more like a notary public or master/mistress-of-ceremony than as someone designated by some external entity to perform weddings.
For religious purposes, consult your entity, or their designee.
[0] currently contested in Rowan County, Kentucky, sigh.
If you're going to quote the First Amendment, it's worth knowing what it says. An "establishment of religion" is a state church, like the Church of England -- the example the framers had in mind, the cause of lots of settlement in the colonies, and an institution that the British monarch is still head of today. Half of the House of Lords are "Lords Spiritual." The coronation ceremony involves an Anglican mass. And it's more than just ritual: as late as 1936, Edward VIII abdicated the crown because he wanted to marry a divorcee, and it was inappropriate for the leader of the Church of England to go against its teachings.
Or take Finland, where the two state churches (the Evangelical Lutheran one and the Orthodox one) facilitate government records of population, in the equivalent of the census, and members of either church (you have to resign membership by a legal process) and also corporations pay a church tax. Or Denmark, whose constitution states, "The Evangelical Lutheran Church shall be the Established Church of Denmark, and as such shall be supported by the State." Or any number of other European countries with church taxes or established churches -- not to mention other parts of the world, such as the Islamic republics.
The state authorizing religious ministers of scores of denominations to officiate marriages in addition to justices of the peace is not, in any sense, an "establishment of religion."
Let's consider that a state (nation-state or US state) might recognize 0, 1, or N religions.
There are many nation-states with 1 or N state-recognized religions, as you point out, but a US state that recognizes any number of religions other than 0 is effectively, in many senses, establishing that set of religions, and excluding any others.
Officiating at weddings is a very light-weight form of recognition compared to tax exemptions, or to religious requirements to hold public office.
My preference for the US would be to keep government out of religion, and religion out of government, keeping that number at 0.
No, I'm taking about state religions, not merely state-recognized religions. These are different. There's only one established religion in the UK, the Church of England, but it is absolutely possible to have your marriage officiated by a non-Anglican or even non-Christian minister. It would be completely wrong to claim that the Roman Catholic Church has been an established church in the UK ever since the Marriage Act 1836 allowed its priests to officiate marriages, or that the CoE is now disestablished because the CoE doesn't permit gay marriage but gay couples can get legally married in other churches. Finland is another example; there are exactly two state churches but dozens of "registered" churches where one may marry.
There is, of course, an extension to be read into that part of the Constitution about not favoring a church in governance; it's sort of the same way the Constitution has an unwritten right to privacy. But you can only read it in so far. It is not a constitutional prohibition on the recognition of churches, just on the establishment of churches. That word had a very specific meaning separate from "recognition" in 1789; it is a testament to the success of that amendment that we in the US have forgotten what it means.
It is worth noting that while everyone who wrote that amendment was still around and deeply involved, Congress wrote this resolution (http://memory.loc.gov/ll/llac/001/0000/00160025.gif):
"Resolved, that after the oath shall have been administered to the President, he, attended by the Vice President, and members of the Senate, and House of Representatives, proceed to St. Paul's Chapel, to hear divine service, to be performed by the Chaplain of Congress already appointed."
And nobody thought that respected an establishment of religion.
(This is not to say, btw, that I personally agree with all of our current policies. I'd very much like to see us further disentangle legal marriage from religious marriage. I just don't think we have anything close to a Constitutional obligation to do so, and we should do so because we the people want to. Being fortunate enough not to have an established church, we also have no Constitutional obligation to stick with the present system, either.)
https://en.wikipedia.org/wiki/Incorporation_of_the_Bill_of_R...
and
http://leg.wa.gov/LawsAndAgencyRules/pages/constitution.aspx