The problem with respect to applications of the abelianization is that the abelian categories one produces are almost uniformly horrible. More precisely they are just too big to deal. So using them to produce results which aren't formal statements about some class of triangulated categories seems like it would be very challenging.
As a concrete(ish) example suppose that $R$ is a discrete valuation ring and let $D(R)$ be the unbounded derived category of $R$-modules. Then the abelianization $A(D(R))$ is not well-copowered - the image of the stalk complex $R$ in degree zero in $A(D(R))$ has a proper class of quotient objects (the reference for this is Appendix C of Neeman's book Triangulated Categories). Since in this case $R$-Mod is hereditary the derived category is really pretty good - we understand the compacts $D^b(R-mod)$ very well and we know all of the localizing and smashing subcategories of $D(R)$. It even comes with a natural tensor product making it rigidly compactly generated and it has a DG-enhancement. On the other hand the abelianization is kind of crazy.
One can take approximations of the abelianization by abelian categories which are more managable. However, I still don't know of any specific applications that aren't just shadows of facts which work for all sufficiently nice triangulated categories.