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" as a 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.