It may be necessary here to distinguish between separating from a spouse while remaining married, and outright divorce - and that for adultery on the part of the one being divorced by the other. As your question shows, Paul was concerned that a couple "marry only in the Lord" (1 Corinthians 7:39).
Paul stated that it wasn't his opinion, but that the Lord required this staying together in a marriage. However, in the next breath, he went on to say "But if she does..." separate, then she cannot remarry. She remains married to the husband from whom she chose to separate. Thus the possibility of being reconciled remains, for marriage is a sacred union.
This shows that Paul was not just concerned with theory, but with practice, and he recognised that separation did happen. This was counsel as to how to retrieve a broken-down marriage. 'Don't divorce. Try to be reconciled.'
What Jesus said does, indeed, show that adultery on the part of a spouse was grounds for divorce. Leviticus 20:10 even states that a man committing adultery should be put to death. But where the adulterer lives, it could be that the spouse wronged against may choose to remain married to the unfaithful partner. Even with adultery, divorce is not the only course. But adultery is, indeed, grounds for divorce, for adultery breaks the marriage bond / union. Death does, also (1 Corinthians 7:39).
When Jesus spoke on divorce, he was addressing a sorry state of affairs that had developed among the Jews. As Matthew Henry's Commentary explains:
"Whoever shall put away his wife, let him give her a bill of divorce"
[Deuteronomy 24:1]: let him not think to do it by word of mouth, when
he is in a passion; but let him do it deliberately, by a legal
instrument in writing, attested by witnesses. If he will dissolve the
matrimonial bond, let him do it solemnly.
"Thus the law had prevented rash and hasty divorces; and perhaps at
first, when writing was not so common among the Jews, that made
divorces rare things; but in process of time it became very common,
and this direction of how to do it, when there was just cause for it,
was construed into a permission of it for any cause (Mat. 19:3).
"How this matter was rectified and amended by our Saviour: He
reduced the ordinance of marriage to its primitive institution: They
two shall be one flesh, not to be easily separated, and therefore
divorce is not to be allowed, except in case of adultery, which breaks
the marriage covenant; but he that puts away his wife upon any other
pretence, causeth her to commit adultery; and him also that shall
marry her when she is thus divorced. Note. Those who lead others into
temptation to sin, or leave them in it, or expose them to it, make
themselves guilty of their sin, and will be accountable for it. This
is one way of being partaker with adulterers Psalm 50:18. (p 1297
Only adultery and death end the marriage union. Thus, a spouse who never committed adultery, but whose partner divorced them anyway, still remains married to that spouse in God's eyes. Indeed, the marriage bond remains between both of them, no matter if one gets a secular divorce. That is why adultery will be committed if there is re-marriage to another, thereafter. But a spouse committing adultery has broken the marriage union and can be divorced, but cannot re-marry. Only the faithful spouse is free to remarry, for the adulterous spouse ended their union.
This isn't about "what Paul allows" - the entire biblical concept of marriage comes together to show just how serious a thing it is to divorce for any reason other than adultery.