History of acid-base reactions
There's a bit of a misconception on what actually happens in the video linked in your question: 

The reactions between alkali metals, namely lithium ($\ce{Li}$) and sodium  ($\ce{Na}$) and water, diluted acid and concentrated sulfuric acid are not neutralisations, but redox reactions, that is transfer of an electron from one reaction partner to the other.

\begin{align}
\ce{2H2O &amp;&lt;=&gt; 2H+ + 2OH-\\
2Na &amp;-&gt; 2Na+ + 2e-\\
2H+ + 2e- &amp;-&gt; H2 ^}
\end{align}

If you take this three reactions as a (mathematical) equation system, with which you're probably more familiar with, and sum it up, the total reaction is
\[\ce{2Na + 2H2O -> 2NaOH + 2 H2 ^} \]
To put this into words:


The metal is dissolved and forms metal hydroxide. 
The resilting solution is alkaline (basic), hence the name alkali metals.
The reaction releases a lot of heat! 
Hydrogen gas is formed.


Since the reaction releases a lot of heat, the hydrogen gas is burnt with the oxygen from the air, yielding water, This is the explosion that you noticed.

Taken these vigorous reactions into account, it is not astonishing that in nature, sodium and and lithium do not exist as metals, but in the form of various salts. It wasn't until 1807 that sodium metal was actually made in the laboratory, followed by lithium in 1818.

Turning back to your question on neutralisations, let's define what it actually means. 

(I apologize if I'm getting to colloquial here!) 

The term neutralisation is typically used to describe that two effects don't sum up to result in something stronger but rather incapacitate each other. In the case of acid-base reactions, this translates to:

\[\ce{effective\ acid + effective\ base -> something\ ineffective + water}\]

In water, a typical example would be the reaction of of a strong acid (such as hyrochloric acid) with equimolar amounts of strong base (such as sodium hydroxide):

\[\ce{HCl + NaOH -> NaCl + H2O}\] 

Adding the right amount of a strong and corrosive acid to a solution of also corrosive and harmful caustic soda a harmless solution of sodium chloride (table salt) in water.
The only side effect of the reaction is that the solution can get pretty hot, but an explosion is not to be expected here.

Typically, usable explosions do not result from neutralisation reactions but from fast combustion reactions in with large volumes of gases are formed, possibly (but not necessarily) accompanied by the release of heat.

Taking into account that strong acids had to be made by men, I'd be surprised to find that these materials were wasted in simple neutralisation reactions.

You will however find historical accounts on the use of the acids or the bases. Nitric acid was used to separate silver from gold (Scheidewasser). Calcium oxide (burnt lime), the precursor of calcium hydroxide e.g. is used to make cement.