I'm not sure I'd summarize it this way:
So, the important point is: not everything happens for a reason.
For a start, the sutta is talking explicitly about "everything a person feels or experiences" -- paṭisaṁvedeti -- maybe that's more precise as a topic than "everything".
Perhaps more importantly it doesn't say that some things "do not happen for a reason" -- I think it suggests that sometimes the reason is "the weather" and so on, just that the reason is not "past actions" i.e. kamma.
And the term it uses for "reason", or "is based" (i.e. "basis"), is samuṭṭhāna -- which I presume is related to paṭiccasamuppāda -- so there's that theory too (which perhaps you're familiar with and already accept), of the "twelve nidanas", which are not exactly kamma; and if you were to say, "ah but kamma is the basis of everything else", there's a different theory i.e. the root kilesas which are said to be the centre of that wheel.
And I think of kamma as personal -- I am the owner of my actions (kamma), heir to my actions, born of my actions. And I think that's a useful theory -- it encourages Right Effort instead of nihilism -- and perhaps it encourages you not to blame other people for "your problems".
But there's the Anattalakkhaṇasutta -- which I think argues that form, feeling, and so on are "not-self" because you can't "compel" (or 'control') them.
It also says in AN 4.77 that the results of kamma are "unthinkable".
In summary I think this leads to the doctrine of emptiness or sunyata. Kamma is a theory or view, a way of seeings things, like a lens or a perspective. And it's a useful one sometimes and (so) it would be wrong to say there's no such thing -- but it isn't the only "reason", the only theory about why things happen or about the 'basis' of everything.