I expected we would agree there.
Though ... those are more along the lines of private prayers, which can be offered for anyone
The actual service within the Church itself - can only take place for those who belong to the Church, and in some sense are in good standing. I'm not sure of ALL the details. I know for instance that suicide can be a problem, since one cannot have repented of it. (However, this is not an absolute - for example, a suicide victim can have a Church funeral if there is reason to believe they were mentally in a state that made them not clearly responsible for their choices ... And suicide is almost an indication of this.) The intention is basically to reserve a Church funeral for those who have not absolutely rejected the Church. Going back to the understanding that it's not a "choice of denomination" as modern Protestants face, but stems from a question historically of "do you consider yourself a part of Christ's Church or not?"
So I'm not sure that our funeral service is meant to "change God's mind" or "get someone into heaven" the way it might seem. We pray for "the servant of God" which is a title bestowed at Baptism and indicates entry into the Church. Though a Church funeral service can also be given to a catechumen who dies, on the basis of their intention and desire to join the Church, as a result of their faith.
Essentially, it seems we are praying for merciful judgment (who knows what unconfessed sins a person may have?) for Christians who die. But I'd have to verify this.