Great Question. I think I am one of the only ones who might be able to answer this. First, we just have to belive in God's Word, and it means what it says. I talked about this with Dr. David Instone-Brewer for a few years. He was very gracious to me. To answer your question on the woman initiating divorce. Ex. 21:10-11 says you are not to neglect your "slave wife" on food, clothing, or marriage rights (love). The Orthodox Jews seem to believe, as well as Dr. David Instone-Brewer, that there was the duties and responsibilities of a husband. Which seems logical. You can't just beat your wife or treat her as a slave without some sort of push back from the Jewish community. David believes, if the slave wife was afforded those requirements, then the free wife would also. And I would agree, in part. Yes, the man had a duty towards his wife, and scripture made the point that the man was not to neglect the basic elements for life and marriage.
But, to answer your question. NO. The Bible does not allow the woman to divorce. An important question to ask is, why was the woman not allowed to divorce her husband in the Old Testament. And once you have that answer, to ask, did that change in the New Testament. Or, is there a teaching by Jesus or His Disciples that say, women are allowed to divorce. The answer is NO. So, as you said, if it does not say it, we should not just assume women could divorce for adultery. Historically, women in Rome were able to divorce their husbands because they claimed to remain under their father's authority when married. And, that makes sense because why else would Mark tell women not to divorce and remarry (Mark 10:12). Josephus has some writings on a woman divorcing her husband (which led to the story of Harod having his brother's wife).
I studied contract law for a while. Wrote a book on it. The Jews believe marriage was a unilateral contract through the Bride Price. Acceptantce through performance, not promise for promise. The woman was bound to her husband because she was under the rule over her husband (Gen. 3:16, Romans 7:1-2, 1 Cor. 7:39). The woman was under the law of her husband. However, the Jews believe a woman could remarry if she was put away by her husband (Deut. 24:1). In contract law, that seems logical. The contract was terminated by the breach of agreement (covenant). Although covenants were made for life, unlike a contract, Moses made a stipulation that allowed all men to put away a wife (out of the home) for a violation against her "sexual performance" (what made her one flesh). Unilateral contracts (agreements) are usually unconditional agreements (promise for performance) since the woman can't breach her promise, however, unilateral contracts do have minor conditions in them. We see this in insurance contracts (unilateral contracts that have limitations and stipulations).
So, NO. The woman is not allowed to divorce her husband because just as the woman was forbidden in the Old Testament (authority) is the same rule (Romans 7:2) that she still falls under. We in America and Western cultures usually marry and make bilateral contracts that give the impression that either party can divorce for breach of contract, but marriage was never meant to be bilateral (woman given away by her father) Numbers 30:2-16. Matter of fact, under Coverture law at common law (starting 1650), women were mostly not allowed to divorce in England or the United States until after 1850. The Matrimonial Clause Act (1857) took the matrimonial matters away from the Ecclesiastical courts and gave it to their secular courts (Courts of Equity), allowing women to divorce for the first time (for adultery compounded by a second offence). This leaked over to America and by the 14th Amendment women used the equal clause to dismantle coverture and many other laws making it possible to work outside the home, own land, and initiate divorce (children were viewed as property of the father). So, women being allowed to divorce by the Church is also a fairly new invention.
I hope that helps. Please fell free to write.