The OED (paywall) notes (besides the definitions relating to "flowing from") a couple of definitions--matters to be decided or matters in contention (matters to be decided by debate or discussion).
16. orig. U.S. Chiefly in pl., and often with modifying word.
a. orig. Psychol. A emotional or psychological difficulty or problem;
a point of emotional conflict.See also to have an issue with at
1977 J. S. Horewitz Family Therapy & Transactional Anal.
268 I think that my own personality and my issues do affect... how I
am as a therapist.
The 1977 is the first quotation under this definition; there are several others.
About the same time, there's this definition:
b. A problem or difficulty with a service or facility; a failing in any system, esp. regarded as a matter to be resolved.
1978 SIAM Jrnl. Appl. Math. 35 233 Immediate dispatch is
appropriate to minimize average wait for all..passengers [on the
shuttle]. There are some technical issues..but they cause no trouble.
The cross-reference to "to have issues with" is listed as U.S. in origin and colloquial:
g. colloq. (orig. U.S.). to have an issue with: to have an objection to; to be unable to accept (a particular condition or circumstance).
1978 Proc. United Glass & Ceramic Workers N. Amer. 101/1 Mr
Chairman, I am going to search my notes. If that is the way it is, I
don't have an issue with that.
To summarize, the OED begins picking up this use of "issue" to mean "problem" in the late 1970s. The legal definition, mentioned in the comment by @StoneyB, appears in the OED definitions with citations going back to the 15th century and seems to me to be related to later use of the term as a matter to be debated or decided, and is related to its use in terms such as racial issue, social issue, and the like.