In general, I don't know of any Anglicans or Lutherans who go to countries like Russia or Greece and try to convert Orthodox persons to their variety of Christianity. Anglicans have exactly three parishes in all of Russia. Since Anglicans consider themselves a National form of Catholicism, and classify Russian Orthodoxy as such, converting them would be considered stupid.
Lutherans also have a similar, albeit somewhat different take on the matter. The Scandinavian Churches hold a VERY similar view as the Church of England does. The rest of us Lutherans might have individual objections to Catholic and/or Orthodox theology (allow me to specify I personally have no objection to Orthodox theology as of yet because I don't KNOW enough of it), but that wouldn't go so far as to call them non-Christian. The idea of going to,for example, Russia, solely for the purpose of converting Russian Orthodox persons, would be beyond our comprehension. There ARE some Lutherans in Russia. But these tend to be people who ALREADY had an interest in Lutheranism, and simply expressed that interest to us, usually after a stay in Estonia or Latvia, both of which states are predominantly Lutheran.
Now, as for OTHER American Protestants (and I point out that Lutheran Christianity is predominantly worldwide and was the first form of Protestantism to exist in 1517 in Germany, quickly moving to Scandinavia and parts of Poland as well), I would agree, they LOVE to poach in the flocks of other Churches. That INCLUDES Lutheran Churches as well as Catholic and Orthodox Churches.
When it comes to the Assemblies of God and other Pentecostal-type traditions, Baptist traditions, and the like, who ARE Christian insofar as they recognise the Trinity, they are just offensive inasmuch as they reject the traditional Churches in favour of their own novel Fundamentalist theological standpoints.
And of course you have your Mormons and your Jehovah's Witnesses, who do not even fit the category of Christian in the traditional sense, since they both reject the Triune God as understood by the Church Fathers. And they LOVE to consider themselves the ONLY true Christians, and they LOVE to preach their heretical doctrines in lands wherein Catholicism, Orthodoxy, or even Anglicanism or Lutheranism are dominant. So you see, we are as much victims of that crap as you are.
I can tell you that in my State in the United States, Lutheranism is so predominant as to be the de facto State Church. NO ONE questions the dominance that it has here. It is largely divided into the Evangelical Lutheran Church in America, Lutheran Church Missouri Synod, Wisconsin Evangelical Lutheran Synod, and Lutheran Congregations in Mission for Christ. The last two are quite small. The first two are quite large, having 4 million members nationwide in the case of the first, and in the case of the second, 2.3 million.
The second largest grouping in this State is the Roman Catholic Church. And again, no one questions this. But here, both the Lutherans of all varieties and the Catholics have the money and the manpower to hold out against all comers.
I realise that in Russia of the '90's, and even to a point now, the religious matters regarding relations between Eastern Orthodox and Eastern Catholic have been and are quite contentious. And where contention exists, an easy hole in the fence gets made for these less, shall we say, desirable of groups, like the Mormons and the JWs to get in.
Furthermore, I can TOTALLY see how it would vex you to no end! I would be p*ssed if that happened to my State here!