How are neutral amines effective bases in organic chemistry? I have found it to be rather strange that neutral amines, such as triethylamine  or DIPEA, find applications in organic chemistry in deprotonating alcohols or even α-carbonyl compounds. Amines are basic but they aren't really that basic, compared to negatively-charged hydroxides or alkoxides. According to Wikipedia, the $\mathrm{p}K_\mathrm{a}$ (in water) of the conjugate acid of triethylamine is $10.75.$ The same values for alcohols are in the range of $16 - 18.$ For water, it is $14.$ Thus, hydroxides and alkoxides are superior alternatives to these amines. 

My motivation for asking about this is that I recently encountered a question in my school chemistry course on why the protection of alcohols with $\ce{MOMCl}$ uses large bulky amines to deprotonate the alcohols, rather than using hydroxides. The answer given was that hydroxides would not be able to deprotonate the alcohols to a significant extent as alcohols are not acidic enough. I find this answer very strange and inconsistent with what I know since as shown by the data above, amines are less basic than alkoxides and hydroxides.
There are several interesting aspects to your question.


  How are neutral amines effective bases in organic chemistry?


First of all, recall the definition of a base. Anything which accepts a proton (or donates a pair of electrons) is a base. Organic bases are very strong and corrosive! These bases should not be undermined, even though they may sound weak (and their stench is worse than inorganic bases). Your lye-free oven cleaner has an organic base, an neutral organic amine such as ethanolamine, which dissolves dirty grease stuck in the oven. This molecule has no charge yet it is a strong base, acting just like sodium hydroxide based oven cleaner in terms of performance.


  Thus, hydroxides and alkoxides are superior alternatives to these
  amines.


The second reason to avoid hydroxides and alkoxides is their solubility in organic solvents. Neutal organic amines easily dissolve in toluene and a variety of organic solvents. NaOH or NaH remain as a suspension.


  MOMCl  uses large bulky amines to deprotonate the alcohols, rather
  than using hydroxides


Again you have to think about the reaction solvent. Is an inorganic hydroxide soluble in your solvent? Then comes an important point. Why use bulky amines? Their sole purpose is to act as a proton sponge in the reaction mixture and prevent the formation of free acid. Bulky amines are preferred so that they don't react easily with the alkyl chlorides or acyl chlorides, thus prevent the formation of their own undesired side products.