I don’t think there is any comparison.
The Rosary is not a part of the Mass, so it really doesn’t matter if a Catholic has a Rosary during Mass. (And as others have pointed out, it’s not the beads that are important, it’s the prayer, which can be prayed without a string of beads.)
But the entire point of a non-Catholic ecclesial community (AKA Protestant) worship service is to worship God through STUDY OF THE BIBLE. The sermon, assuming that the pastor isn’t a “liberal,” is supposed to be an exposition on the Word of God, the Holy Bible.
Even the SONGS are supposed to be Biblically-based and often the musician will cite a Bible passage. And if someone stands and gives their personal testimony, they will often cite a Bible verse or passage and ask the congregation to please look it up in their Bibles.
When I was evangelical Protestant, I and everyone else, including children, brought our Bible to church and turned to the Scripture passages during the readings. Then during the sermon, which was about the Scripture readings, we followed along in our Bibles.
Often the pastor would ask us to mark a certain passage in our Bibles, and sometimes, he/she would ask us to write a little note alongside of the passage. My old NASB is full of such notes and underscores. (It makes it very easy for me to find passages of Scripture even without a concordance.)
Many of us also took notes during the sermons. Many times, pastors will cite other Bible verses and passages in his/her sermon, and we would write down the references to look up later.
And most importantly, we would STUDY those notes and the Scripture passages throughout the week and try to incorporate them into our lives.
Many members of non-Catholic ecclesial communities (AKA Protestant) take the Scriptural injunction to “examine all things” very, VERY seriously. They bring their Bibles, sometimes multi-translation Bibles (and even Greek versions) to “keep watch” on what the pastor is saying. They don’t want the pastor to feed them anything that isn’t really in the Bible. So they check up on him/her by following carefully in their Bibles.
I did this for almost a year in the Catholic Church out of habit. I have only recently begun taking my Bible into Mass and reading out of it during the Readings. A Catholic priest at a Catholic Family Conference urged Catholics to carry their Bibles to Church, because the Catholic Church is a BIBLICAL church. Sounds good to me.
IMO, asking a member of a non-Catholic ecclesial community (AKA Protestant) to not bring their Bible to church would be like asking them to just skip church. The whole point is Bible study. Without the Bible, you are studying only the words of a man/woman.