Why does hydrogen so easily take the place of oxygen in organic molecules during reduction reactions?
But, why does it generally follow that hydrogen takes the place of an oxygen molecule during a hydrogenation/reduction reaction [...]


Ok, if at all, it's an oxygen atom ;)

I don't see that general trend towards complete deoxygenation, neither in catalytic hydrogenations, nor in hydride transfer reactions.

It usually takes


rather drastical conditions 
quite some special reagents
a particular type of derivatisation 


or a combination of any of these to convert oxygen-containing compounds to alkanes.

Let's have a look at some examples:

$\ce{R1-CO-R2 -&gt; R1-CH2-R2}$


Ketones and aldehydes may be directly reduced to alkanes in the Clemmensen reduction. The "mildest" variants use anhydrous $\ce{HCl}$ and amalgamated $\ce{Zn}$ in organic solvents.
The clasical Wolff-Kishner reduction, using hydrazine and sodium in diglyme, involves the in situ functionalization of the carbonyl and isn't a prototype for a mild reaction either. Admittedly, milder variants exist, using N-tert-butyldimethylsilylhydrazone and catalytical amounts of a Lewis acid, e.g. $\ce{Sc(OTf)3}$
Tosylhydrazones may be converted to alkanes using $\ce{NaCNBH3}$, $\ce{Na(CH3COO)3BH}$, or boranes.
Catalytic hydrogenation over Raney nickel is possible after converting the carbonyl to a dithiane; so it's actually a desulfurization.


$\ce{R1-CH2-OH -&gt; R1-CH3}$

I'm not aware of any transformation that does not involve functionalization.


Alkyl tosylates can be deoxygenated using $\ce{LiEt3BH}$.
Acylation of alkohols with thiocarbonyl chlorides furnish thiocarbonates, which may be deoxygenated under using either tin hydrides ($\ce{R3SnH}$, Barton deoxygenation) or $\ce{B(CH3)3}$ in water/benzene.
Alkanes may also be obtained from alkohols under Mitsunobu conditions ($\ce{Ph3P}$, $\ce{DEAD}$, o-nitrobenzenesulfonylhydrazine).


$\ce{R1-CH2-COOH -&gt; R1-CH3}$

Direct transformations of carboxylic acids to alkanes typically involve decarboxylation, i.e. the loss of $\ce{CO2}$. This may be achieved either by the Barton decarboxylation, or via photoinduced electron transfer reactions with electronically excited (electron) acceptors. In the course of the latter reactions, a carboxylate is oxidized to an acyloxy radical, which then undergoes decarboxylation.

To sum it up: It's not that easy ;)