In Modern Japanese, the dictionary form of a verb is a single surface form, but it represents the evolution of two historical forms, the 終止形 and 連体形. Although it's often convenient to treat these as a single form, I think it's still useful to make the distinction sometimes, and this may be one of those times.
In Classical Japanese, the 連体形 was able to function as a noun. In fact, that's how we ended up with the particle
が marking a subject; it was originally the genitive
の) still seen in
我が and names like
青木ヶ原, and it linked the subject (a noun) with the nominalized form of the verb (the 連体形, another noun!). That's also how we got the sentence-final
が meaning "but"; it was able to attach after a verb because that verb was nominalized!
In Modern Japanese, the 終止形 was lost, but the function remains, so we say that the 終止形 has the same form as the 連体形. (With the exception of だ, of course.) Perhaps because of this change, the 連体形 of a verb doesn't generally function as a nominalized form, so an overt nominalizer such as
こと is usually inserted. And just as the function of the 連体形 has been reanalyzed, so have the roles of the particle
However, in this case, it seems that
目指す is functioning as a noun, even though the dialogue is clearly Modern Japanese! And since this kind of construction is accepted in modern published sources, I think we can conclude that using
目指す here as a noun must be grammatical. So either this form is able to function as a noun, or there is a zero marker turning it into one. Here's how I'd sum up both analyses:
- We can say that 目指す is not the 終止形 but the 連体形, and that this form can still function as a noun, although it's not as common anymore.
- If we reject that the 連体形 can function directly as a noun in modern Japanese (or at all), then we can say there is a zero-nominalizer after the verb, similar to
の but perhaps not equivalent. Here is a question about zero nominalization in Japanese.
For more details about the historical development of
の, please see Shibatani's The Languages of Japan, pp.347-357.