I am no specialist of Szasz, but it seems that you are raising both a philosophical (ethical) and a practical (judicial) question.
A priori, I see a weakness in this sentence:
It seems only reasonable to claim that. It seems to me that saying otherwise is close to the claim that we are responsible for everything bad that will ever happen to anyone at all, as much as it is to the claim that we are responsible for some things.
That could be a fallacy of the "excluded middle" since practical experience is complicated and things cannot be open and shut. There is no "quantum shift" (or "slipping") from one absolute proposition to the opposed one.
In more detail, a key distinction exists between responsibility and liability. Responsibility is a much broader concept that covers present (doing one's intended or expected actions), future (predicting, preparing, etc.), and past (assuming causality and consequences of one's actions). When one introduces fault (and hence the liability of a person to repair the damage), then a number of logical pre-requisites are typically required [philosophers of law or law experts do not hesitate to complete or amend]:
- Something adverse affected some party B in the past, against its own will (hence there has to be a damage or an objective prohibited action; plus see the maxim volenti non fit iniuria, i.e. if the other party was willing and knowing in the first place, there might be no complaint possible).
- A causal effect was established between B (victim) and a person A (defendent).
- The action of A violated the expected behavior in such circumstances, either prescriptive (do) or prohibitive (do not).
Fault thus appears in a social context. Hence the question whether "we are responsible for everything that will ever happen to anyone at all" is far too general and we do not need to answer it in order to answer the original question (it belongs to Occam's Razor).
The relevant question about liability is establishing an objective (observable, documented, etc.) causality link (there cannot be fault without it), or at least a "faulty" action/inaction. Furthermore, one could directly cause a damage to someone else, without being considered liable: typically a person who falls from a stair in mall, hurting another one, can be held innocent of the injury, in the case the owner failed to apply a prescribed safety measure that would have prevented the damage, in which case the owner is considered at fault).
It is correct to also frame the question from a subjective perspective of "cognitive", i.e. what one can be cognizant about, but as a subordinate question. In judicial matters, that is of course very relevant, together with another key aspect: intention. This applies to civil and criminal law, and more generally to adjudications according to the accepted morals of a human group.
Note, however, that they come second in the adjudication (on the part of a judge, peers) as extenuating or aggravating circumstances. Hence ignorance or inability of seeing the consequences of one's actions can be invoked, but it is subordinated to the establishment of an objective (observable, demonstrable) causality as well as the violation of rules of behaviour.
Hence an adjudication on subjective factors removes neither the objective causality nor the violation of the rules (e.g. if a mentally irresponsible person killed another, that person still killed the victim and that is still considered inacceptable).
That is from the perspective of society. From a personal perspective, person B (victim) could clearly chose or not chose to take more responsibility for their own actions (e.g. the person who got hurt in the stairs could have looked around while walking, instead of at their mobile phone; and they might draw a lesson for the future, on the danger of it). But as long as any legal/moral obligation (prescriptive or prohibitive) was not violated, the rest of society society will not interfere. And that reconnects with the political philosophy of the social contract: that the aim of the rule of law is not to interfere with personal behaviours, but to provide the largest individual freedom that is reasonably achievable in an organized society.