Logic, paraconsistent or not, does not exactly make something happen, it is applied to reshuffle information already contained in a system. Paraconsistent logic does not even have to be applied to inconsistent systems, and even when it is, derivable contradictions do not have to be interpreted as "true".
What we need is not logic but semantics, although what kind of logic is used does impose some constraints on how the resulting system can be interpreted. Semantic interpretations that admit true contradictions, a.k.a. dialetheias, are called dialetheist. Priest and Routley, the founders of dialetheism, did draw inspiration for their interpretation of naive set theory from Wittgenstein’s remarks about the Russell’s paradox:
"Why should Russell’s contradiction not be conceived of as something supra-propositional, something that towers above the propositions and looks in both directions like a Janus head? The proposition that contradicts itself would stand like a monument (with a Janus head) over the propositions of logic".
This was developed into a body of inconsistent mathematics, by Meyer and others. The point is to obviate the negative conclusions of Gödel's incompleteness theorem by rejecting one of its premises, the assumption of consistency. Meyer's inconsistent arithmetic R# has no undecidable statements and he proved by finitary means that contradictions within it do not affect any numerical calculations. This is in a sense a realization of Hilbert's programme of proving consistency of arithmetic by finitary means, or at least as close as one can come.
Similarly, dialetheist interpretations were used to deal with the semantic paradoxes, like the famous Liar. If we admit true contradictions then a resolution of the Liar would be that the "I am false" sentence is just that. Hegel's philosophy, with its dialectic, and other non-dualist systems with their "unity of the opposites" (neoplatonism, Buddhism, Advaita, etc.) arguably affirm dialetheias, although this is debatable. Hegel does say that "one of the fundamental prejudices of logic as hitherto understood... [is that] the contradictory cannot be imagined or thought", but he uses "logic" in a different, old, sense, closer to today's "epistemology".
However, dialetheism is not the only, and not even the most common, way of admitting "impossible things". Pace Hume, who thought that things impossible can not be believed, or even conceived, things that turn out to be impossible are routinely conceived in reasoning provisionally, for example in reductio arguments. An ancient example is Euclid's proof which considers a rational number whose square is 2, and after a series of manipulations concludes that such a number does not exist after all, because a contradiction results. Russell's set is treated the same way in his paradox. To this day we do not know if an odd perfect number (equal to the sum of its proper divisors) is impossible or not, but mathematicians have been proving things about them for centuries. In other words, one need not believe in true contradictions to have a need to reason about the impossible.
This is handled by the epistemic logic, logic of what is known. Since the knower may not be smart enough to see through all the consequences of her assumptions she may well believe some hidden contradictions. Such belief systems are modeled using modal semantics that in addition to possible worlds admits impossible worlds. The sets of sentences describing them can imply contradictions, but derivations of contradictions have to be "long". The abridged descriptions are not closed under the logical consequence and hence avoid "overt" inconsistency. Other dialetheist and non-dialetheist interpretations are equally possible, as Priest points out:
"As far as I can see, any of the main theories concerning the nature of possible worlds can be applied equally to impossible worlds: they are existent nonactual entities; they are nonexistent objects; they are constructions out of properties and other universals; they are just certain sets of sentences."
Non-existent objects in ontology predate even modal logic, they were proposed by Meinong already in 19th century.