2. Question: Why can’t Omer be written down to fulfill the mitzvah even according to the view that writing is like speaking?
A. The Mishna Berura (47:4) brings an argument by Torah learning whether writing is like speaking or thinking the question being whether you have to say the blessing for learning Torah before writing Torah thoughts. On the one hand it’s an action so it’s like speaking on the other hand he didn’t say anything and the blessing is for speaking in learning. The Mishna Berura concludes that one should be strict and not say a blessing in the morning upon writing Torah notes until he is ready to verbalize them.
B. The Aruch Hashulchan says that by Torah learning the purpose of writing is to reveal the thoughts in one’s mind and that is why one opinion says writing is like speaking. Why then would writing the omer be different?
Answer: The mitzvah by omer, krish Shema, and the amida is speaking it out so writing isn’t enough. So if you did it you would have to say the blessing over again and recite the omer verbally. (See Dirshu Mishna Berura 489 footnote 43.)