MaTeX doesn't change fonts on axes?

The first block of code you show,

BarChart[{-10, -5, 10, 50}, AxesLabel -> {"X-Axis", "Y-Axis"}, 
 ChartLabels -> {"1", "2", "3", "4"}, ChartStyle -> {"Pastel"}, 
 BaseStyle -> {FontFamily -> "Latin Modern Roman", FontSize -> 16}]

does not use MaTeX and does not need MaTeX. All you need is to install the Latin Modern Roman font, and specify it with FontFamily.

Where this gets tricky is that the name you must use with FontFamily is different for each operating system. I use "Latin Modern Roman" in the MaTeX documentation because that's the correct form on OS X (my platform of choice). However, on Windows it is called something different. "LM Roman 12" if memory serves.

Please go to Format -> Show Fonts to reveal the font list, and check the correct name of the font on your operating system.

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Another note: You are linking to MaTeX's website, but the bulk of the documentation is part of the package, and can be accessed through the Documentation Center (Mathematica's Help menu). The MaTeX tutorial included in the main documentation does mention this problem:

I am not sure if you want to change Axes labels to Latex only or also change the ticks. According to MaTeX web site, to also change ticks to Latex, one needs to manually typeset them to Latex.

I do not know why texStyle is not making this change automatically.

But this works. It means you have to type set them manually using MaTeX command on each item you want to show as Latex

<< MaTeX`

lb1 = MaTeX["\\text{X-Axis}", Magnification -> 1.25];
lb2 = MaTeX["\\text{Y-Axis}", Magnification -> 1.25];

BarChart[{-10, -5, 10, 50}, 
 AxesLabel -> {lb1, lb2},
 ChartLabels -> {MaTeX[#, Magnification -> 1.3] & /@ {"1", "2", "3", "4"}}, 
 ChartStyle -> {"Pastel"},
 PlotRangePadding -> Scaled[.05]
]

Although for some reason, the font is not sharp.