We study the fate of the surface states of Bi$_2$Se$_3$ under disorder with strength larger than the bulk gap, caused by neon sputtering and nonmagnetic adsorbates. We find that neon sputtering introduces strong but dilute defects, which can be modeled by a unitary impurity distribution, whereas adsorbates, such as water vapor or carbon monoxide, are best described by Gaussian disorder. Remarkably, these two disorder types have a dramatically different effect on the surface states. Our soft x-ray ARPES measurements combined with numerical simulations show that unitary surface disorder pushes the Dirac state to inward quintuplet layers, burying it below an insulating surface layer. As a consequence, the surface spectral function becomes weaker, but retains its quasiparticle peak. This is in contrast to Gaussian disorder, which smears out the quasiparticle peak completely. At the surface of Bi$_2$Se$_3$, the effects of Gaussian disorder can be reduced by removing surface adsorbates using neon sputtering, which, however, introduces unitary scatterers. Since unitary disorder has a weaker effect than Gaussian disorder, the ARPES signal of the Dirac surface state becomes sharper upon sputtering.
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