In Islam, must a man be the one to make the marriage proposal? If so, why?
It is not necessary for the groom to initiate the proposal, rather it is also possible for the wali of the bride to do so as is evident from the Quran regarding the marriage of Moses عليه السلام:
قال إني أريد أن أنكحك إحدى ابنتي هاتين
He said, "Indeed, I wish to wed you one of these, my two daughters ... "
فمن الحسن عرض الرجل وليته ، والمرأة نفسها على الرجل الصالح ، اقتداء بالسلف الصالح
So it is good for a man to offer his female relative who is under his care in marriage, and for a woman to offer herself in marriage to a righteous man, following the example of the righteous salaf.
The same is also evident from hadith regarding the proposal of Umar to Abu Bakr and Uthman (Bukhari 5122).
And it is possible for the bride herself to convey the proposal:
أن امرأة عرضت نفسها على النبي صلى الله عليه وسلم
A woman presented herself to the Prophet ﷺ for marriage
وفيه استحباب عرض المرأة نفسها على الرجل الصالح ليتزوجها
And in it is the recommendation of a woman proposing herself to a righteous man for the purpose of marriage
Also refer to Ibn Hajar and al-'Ayni's commentary on the hadith, translation available on islamqa.
"Must a man be the one to make the marriage proposal?" . In short. No. In tradition, man would be the one who propose, because he is the one to be burden with the wellbeing of a family.
Anyone can be the first to propose, yet it will come to each individual rights weather to accept it or not.
Marriage in Islam is based on mutual peace, love, and compassion. No one should force a marriage. Further discussions can be found here: https://www.iium.edu.my/deed/articles/genderequityinislam.html
The female has the right to accept or reject marriage proposals. Her consent is prerequisite to the validity of the marital contract according to the Prophet's teaching. It follows that if by "arranged marriage" is meant marrying the girl without her consent, then such a marriage is nullifiable is she so wished.