Slips of Tongue Serve as Strong Evidence
I don't know how far syntax studies have gone, or what their concerns are these days. As far as I've heard, there was a time when it was fashionable to employ Chomsky's transformation-generative model in a thesis or talk, but that was some time in the 1970's-80's ?
To me Chomsky's importance lies in his idea of deep structure that is underlying the surface structure. While the idea of double articulation of sounds in General Linguistics was a good starting point to study the surface structure of morphemes, words, syllables, and phonemes, the properties of the sentence itself remained cryptic in its impenetrable terseness.
By hypothesizing a number of simple sentences for all naturally observed sentences and phrases, and a transformational rule such as deleting redundant parts, all grammatical and ungrammatical speech could be rationally explained.
Slips of tongue occurring at this deep level is sure proof that grammatical entities such as parts of speech, phrase structure rules, and transformational rules exist. This is equally the case for speech errors supporting the existence of phonemes, morphemes, morphophonemic rules, and phonological rules.
In short, our grammatical understanding and our ability to analyze complex, cryptic, and even erroneous sentences has been greatly deepened thanks to Chomsky. By understanding I mean "metalinguistic ability" to understand the structural properties of each sentence produced in a natural language; but it can also help in an L2 setting where a 2nd/foreign language is learned after the critical years.
More on Chomsky, see p. 6 in
Linguistics: The Scientific Study of Human Language, Victoria A. Fromkin,
Introduction
More on speech error, see Fromkin,
Slips of the Tongue, 1973
(Sorry, can't find text either on-line or for sale...will supply asap.)
Victoria Fromkin's studies in speech errors and linguistics general
Obituary