The Lutheran Small Catechism with Explanation (ESV) provides a classic Protestant look at ecclesiology (how one views the Church), but I find it very unconvincing and full of problems. My conclusion is that the Lutheran alternative does not seem plausible, and it most certainly can not disprove the claims of the Church.
1. Under the question, “What is the holy Christian church?”, it answers:
“The holy Christian church is the communion of saints, the total number of those who believe in Christ. All believers in Christ, but only believers, are members of the church (invisible church).”
This is sort of true, but what if someone has faith and still intentionally separates themself from the Church by heresy? For example, are Arians members of the Church? They believe in Christ. Are Mormons also members of the Church? What about Jehovah’s Witnesses? This kind of vague, “invisible” membership leads to all sorts of problems, and it leads to the loss of absolute truth. (See the very varied views of Protestants.)
A single institutional Church is necessary, because some doctrines are “hard to understand” (2 Peter 3:16) and they need to be consistently preserved and articulated.
2. Under the question, “Why do you say ‘I believe’ in the church?”, it answers:
“A. Because faith, which makes people members of the church, is invisible, the church is invisible to human eyes.
B. The Scriptures assure us that the Holy Spirit continues to gather and preserve the church.”
On the second part of this answer, I have no complaints. The Holy Spirit certainly does guide the Church. However, on the first point, it cites Luke 17:20-21 and 2 Timothy 2:19 for support, taking both passages out of context. The first passage actually refers to the “end times” and people wondering about when they will be and what they will entail, and this is made clear by the rest of the chapter. The second passage simply points out that, despite heresy being almost everywhere, “the firm foundation of God stands” and “the Lord knows those who are His”.
The Church is not invisible.
3. This Lutheran Catechism also makes the points that the Church’s “one and only head is Christ” and the Church “belongs to Christ and is built on Him alone”, but this is misleading and an intentional jab at the Church.
Christ is the now-invisible head of the Church, in that He fills Her with grace and protects Her from grave error, but the Church must have a visible head to represent Him: the Vicar (representative) of Christ, the Successor of St. Peter — the Pope.
It is true that only Christ could lay the foundation for His Church (1 Corinthians 3:11) and that He is the cornerstone (Ephesians 2:20), and the Church absolutely recognizes this. He laid the foundation when He appointed Peter as the visible head of the Church (Matthew 16:18-19) and He is still the cornerstone — without Christ, the Church would crumble.
Because only Christ can lay the foundation of a Church, Martin Luther had no authority to start his own sect — unless, of course, there is some sort of evidence that definitively shows that Christ transferred His authority to him. Naturally, this evidence does not exist.
Also, remember that not everyone is “called” to Church leadership (Hebrews 5:1-4).
4. Additionally, this Catechism teaches that “the holy Christian church is to be found where ‘the Gospel is purely taught and the Sacraments are correctly administered’ (Augsburg Confession VII 1)”.
I absolutely agree with this point, because only an organization that distributes the sacraments is a “Church” in the proper sense, though it may not be in communion with the Church. “Christ’s Spirit uses [them] as means of salvation, whose power derives from the fullness of grace and truth that Christ has entrusted to the Catholic Church.” (Catechism of the Catholic Church, #819)
However, even after taking this into account, I also realize that the Lutheran understanding of both the Gospel and the sacraments is distorted.
Lutherans typically believe that there are only two sacraments (Baptism and Communion). Catholics, meanwhile, recognize a total of seven: Baptism, Communion (the Eucharist), Confession (Penance), Confirmation (or Chrismation), Marriage, Anointing of the Sick, and Holy Orders. Lutherans usually think of these other five as rites that do not necessarily contain God’s grace, but are still historically practiced.
Just one example of the Lutheran sacramental problem is that they hold to sacramental union (Christ is “in, with, and under” the bread and wine), while the Church holds to transubstantiation (the bread and wine become the literal Body and Blood of Christ), which is the traditional view. The Lutheran departure from the historical view seems to reveal “a morbid interest in controversial questions and disputes about words” (1 Timothy 6:3-5). Is their emphasis here more important than unity?
Meanwhile, Lutherans also debate over whether or not Confession is a sacrament. Martin Luther said one thing, but the official Defense of the Augsburg Confession says another.
“Nevertheless, it has seemed best to restrict the name of sacrament to such promises as have signs attached to them. The remainder, not being bound to signs, are bare promises. Hence there are, strictly speaking, but two sacraments in the Church of God – baptism and bread; for only in these two do we find both the divinely instituted sign and the promise of forgiveness of sins.” – Martin Luther [link]
“If we call Sacraments rites which have the command of God, and to which the promise of grace has been added, it is easy to decide what are properly Sacraments. For rites instituted by men will not in this way be Sacraments properly so called. For it does not belong to human authority to promise grace. Therefore signs instituted without God’s command are not sure signs of grace, even though they perhaps instruct the rude [children or the uncultivated], or admonish as to something [as a painted cross]. Therefore Baptism, the Lord’s Supper, and Absolution, which is the Sacrament of Repentance, are truly Sacraments. For these rites have God’s command and the promise of grace, which is peculiar to the New Testament.” – Article XIII of the Defense of the Augsburg Confession [link]
With disagreements over the fundamental natures of the sacraments and their generally invalid claims to apostolic succession (which is necessary for the validity of the sacraments), Lutherans do not have a “Church” in the proper sense.
5. Protestant ecclesiology has wrecked the doctrinal and visible unity that God demands.
In Galatians 5:16-21, St. Paul condemns “dissensions” and “factions” as “deeds of the flesh” that will result in the causers “not inherit[ing] the kingdom of God,” and in Romans 16:17, he teaches that Christians should “turn away from” them. Protestants have, unfortunately, disobeyed this command.
Unity is Christ’s prayer for us (John 17:11), so let us become unified again, visibly and invisibly.
“Since Christ suffered for the Church and since the Church is the body of Christ, without doubt the person who divides the Church is convicted of lacerating the body of Christ.” – Council of Florence, Session 9 (23 March 1440) [link]
(All verses are from the NASB translation.)
Martin Luther said and wrote many heretical things, but here are a mere five quotes, along with Biblical passages that disprove them.
1. “For reason is directly opposed to faith. This is why you must let reason go. It must be killed and buried in believers.” 
(Answer: Proverbs 14:15, Colossians 1:9, Colossians 4:5-6, 1 Thessalonians 5:19-22)
2. “..[A Christian] could not lose his salvation, however much he sinned, unless he refused to believe. For no sin can condemn him save unbelief alone.” 
(Answer: Hebrews 10:26-27)
3. “Be a sinner, and let your sins be strong, but let your trust in Christ be stronger, and rejoice in Christ who is the victor over sin, death, and the world. We will commit sins while we are here, for this life is not a place where justice resides. …No sin can separate us from Him, even if we were to kill or commit adultery thousands of times each day.” 
(Answer: Psalm 4:4, Matthew 5:48, Hebrews 10:26-27)
4. “It is good that we have such a man [Jesus Christ], because God in himself is cruel and bad.” 
(Answer: Psalm 51:1, Jeremiah 9:24, Jeremiah 31:3)
5. “I miss more than one thing in this book [Revelation, aka Apocalypse], and this makes me hold it to be neither apostolic nor prophetic…and [I] can nohow detect that the Holy Spirit produced it. …My spirit cannot fit itself into this book.” 
(Answer: Revelation 22:19)
1. Luther’s Sermon on Matthew 19:13-15. “What Luther Says,” 1:485-486 (entry 1440). [link]
3. Let Your Sins Be Strong: A Letter From Luther to Melanchthon. Translated by Erika Bullmann Flores. [link]
5. M. Reu, Luther’s German Bible: An Historical Presentation Together with a collection of Sources (Ohio: The Lutheran Book Concern, 1934) 174-175 [link]