I can see from the wording of your question that you already see the WRONG USE of this verse by church leaders today, for they are, as you say, taking it “out of context.” These words are found in 1 Chronicles 16:22 and Psalm 105:15 and both of those verses are speaking of God’s sovereign CHOICE of Abraham, Isaac, and Jacob as His PROPHETS. I will now quote Psalm 105:6-15 to bring out the “full context” to prove this:

“O seed of ABRAHAM His servant, You children of JACOB, His CHOSEN ONES! He is the LORD our God; His judgments are in all the earth. He remembers His covenant forever, the word which He commanded, for a thousand generations, the COVENANT WHICH HE MADE WITH ABRAHAM, AND HIS OATH TO ISAAC, AND CONFIRMED IT TO JACOB for a stature, to Israel as an everlasting covenant, Saying, ‘To you I will give the land of Canaan as the allotment of your inheritance,’ when they were few in number, indeed very few, and strangers in it. When they went from one nation to another, from one kingdom to another people, He permitted no one to do them wrong; Yes, He REBUKED KINGS FOR THEIR SAKES, saying, ‘Do not touch My anointed ones, and do My prophets no harm.”

The words of King David in this Psalm are crystal-clear; God CHOSE Abraham, Isaac and Jacob as His PROPHETS to declare the covenant He had made with them and no one, including kings, was allowed to touch His ANOINTED ONES. David knew that God is the Sovereign Ruler who chooses men at times to be His prophets and he also knew that God CHOSE KINGS to be “His anointed ones.” You may recall when King Saul was hunting for David with a desire to kill him, David and his men found Saul and his men sleeping and they had the opportunity kill him (1 Samuel 26:1-7). In verses 8-11 we read, “Then Abishai said to David, ‘God has delivered your enemy into your hand this day. Now therefore, please, let me strike him once with the spear, right to the earth; and I will not have to strike him a second time!’ And David said to Abishai, ‘Do not destroy him; for who can stretch out his hand against THE LORD’S ANOINTED, and be guiltless? …As the LORD lives, the LORD shall strike him, or his day shall come to die, or he shall go out to battle and perish. The LORD forbid that I should stretch out my hand against THE LORD’S ANOINTED.” David knew that God had CHOSEN SAUL to be KING and God alone had the right to remove him from power. Saul was a wicked king and in time he died a terrible death (see 1 Samuel chapter 31) which he deserved, but while he lived God protected him. 

We never read the words, “Do not touch my anointed ones; do my prophets no harm” in the New Testament.” As we have seen, they refer to PROPHETS and KINGS that God chose in connection with His earthly people, Israel. For “church leaders” to apply these words to themselves is not only wrong, but it is shameful. As you say, they are using them to “shield themselves from accountability and criticism.” Listen to what we read of “church elders who sin” in 1 Timothy 5:19-20, “Do not receive an accusation against an elder except from two or three witnesses. THOSE WHO ARE SINNING REBUKE IN THE PRESENCE OF ALL, that the rest also may fear.” Their “sin” may be teaching false doctrine, or it may be a case of immorality. This teaches us that they are not immune to God’s discipline and God will use His children to “publicly rebuke” them in front of the whole assembly. Their sin could also be serious enough where they would have to be “publicly removed from the local church” (see 1 Corinthians chapter 5). Though a church leader is not looked upon as “God’s anointed,” he is still in a very responsible position. He is indeed accountable and sin will be dealt with accordingly. James 5:1 speaks to this, “My brethren, let not many of you become teachers, knowing that we shall receive a stricter judgment.”  (DO)  (717.3)