Consider utterances such as
"Ouch, I got something in my eye"
"Something made a noise, and it kept me up all night"
Now, in some circumstances, the identity of the referent is relevant to the further context (as per expectations that speakers learn while acquiring the language). In some circumstances, however, the identity might be entirely irrelevant - tiny particles you get in your eye, things that make noises, etc.
Such things could have a separate pronoun - one that is simply an indefinite, "antitopical" pronoun, which I will call the 'inconsequential'. Reusing the same pronoun in an utterance of roughly "paragraph"-length would indicate it's the same thing being referred to (but that still, it's identity is not all that interesting, though the continuity of its identity is maintained). Switching the referent to a more topical pronoun requires explicitly stating something like 'itinconsequential was [NP], itregular was' or some other construction along those lines.
Now, what other things could we weave around this concept? Maybe indefinite pronouns do not trigger third person congruence on verbs – but since only subjects (or only subjects and objects) have congruence on verbs, the pronoun has existed for other constituents, and been generalized to work as a subject as well. This could well lead to it having some morphological gaps - i.e. no case distinction between subject and object, and even possibly less. Of course, this might indicate that the language usually does have those cases, at least in the pronominal system.
(This of course is only an indication as to what kinds of questions we ask, rather than as to actually meaningful things about the language).
Further though, this might have some use for other purposes - Navajo has a possessive prefix that is somewhat similar in meaning, which is used to permit using inalienably possessed nouns even without specifying a possessor, i.e. shizé'é: my father, azhé'é: someone's father, a father.
So, this could easily extend into that kind of construction as well.
As for getting no congruence on verbs, this could lead to a situation where a noun marked as object gets subject congruence (and the object congruence is omitted altogether), so the language essentially forms passives by
subject: itinconsequential object: Noun, acc Verb: [subj: congruence with Noun]
Having both a case system AND subject and object congruence is not very common, as far as I can tell (some languages of Beringia and northernmost North America excepted), so it might just happen that this is somewhat typologically messed up. Let's instead go for the following solution: itinconsequential and other indefinites lack subject congruence, but not object congruence. In the case of no object congruence being present, the object has been promoted to syntactical subject status (while the indefinite has been demoted to unmarked oblique), whereas indefinite object congruence of course tells us that the subject is the regular noun.
A VP in which one indefinite object congruence is present, as well as an indefinite pronoun - and nothing else - the only interpretation is that the indefinite pronoun is the subject, and the indefinite morpheme on the verb is the object. Thus, "something acts on something" is distinguishable from "itinconsequential acts on something", but "something acts on something" is indistinguishable from "something acts on itinconsequential", and same goes for "itinconsequential acts on something/itinconsequential". The loss of these distinctions does not seem all that worrying, though.