Perhaps this could be better understood by looking at the most popular holiday of the world, specifically Christmas. Christmas is supposedly, or commonly celebrated or a celebration of the birth of the Christ Child, Jesus. Now, let’s find in the Scriptures where God instructs us to celebrate the birth of His Son. There is not one place in either the Old or New Testament telling us to honor our Savior’s birth each year. As a matter of fact, Jesus Himself tells us to remember, His death (Luke 21:14-20; I Corinthians 11:23-26)! Undoubtedly, it is important that He was born, but the fact that He died and how and why He died has farther-reaching and more eternal consequences!
What most people do not fully realize is that Satan has deceived this whole world (Revelation 12:9). Many of the comfortable, familiar customs and traditions of this world have, indeed, been imitated from utter paganism and have nothing to do with proper Christianity. God tells us to prove all things (I Thessalonians 5:21). Before we accept any ritual, we should always inquire into its origins. We must assure ourselves that it does not transgress any of God's laws and that it follows the traditions and practices of the early New Testament church. So, is the sign of the cross really an emblem of true Christianity or is it something far different? Therefore, Christians saying they are saved by the power of the cross is also symbolic of the Power of the Blood of Christ in its inherent purity as it was shed on the cross. As to the shape of, or whether or not the cross had a crossbeam, or was just a stake for torture, is actually irrelevant.
In Romans 1:18-20, Paul declares that things inclusive of God's existence, power, and nature are clearly seen, but men tend to squelch the truth. What God wants man to know; man freely ignores and squelches through the adding of beliefs, customs and traditions that cloak the truth. The truth is still there, hidden behind a screen of falsehoods that most never attempt to remove.
Theologians call this method syncretism. According to Merriam Webster's Collegiate Dictionary, syncretism is "the combination of different forms of belief or practice." Syncretism might perchance term in other arenas, like philosophy, but scholars use it most singularly in religious contexts. Syncretize, the verb form of the word, is very illuminating. It means "to attempt to unite and harmonize especially without critical examination or logical unity." In other words, those who syncretize will often attach one belief or tradition to their religion without trying to determine whether it is correct to do so.
Hope this is of help towards answering the question of why some Protestants hesitate to be accepting of symbols or symbolism when it comes or relates to Godly things. Otherwise outside of the bible I am not aware of any universal Protestant doctrine.
I have no degree, however I did two plus years of biblical studies at Toccoa Falls Bible College in North East Georgia, and I have been born again since March 1985. My answer is from those, and my own studies of scripture and many other text and many websites where I researched. I have tried to be as fluent as I could. I am somewhat novice to putting my words out as such.