What happens if don't use MgCl₂ in buffer solution in Ca²⁺/Mg²⁺ titration?
Summary The Magnesium is added because the color change is easier to spot with it, even though calcium also changes the color of the solution in presence of TBT.

Long explanation and experiment

Even though your question might looks silly at first, I know it makes sense because I work with Soil Analysis, and the titrimetric determination of $\ce{Ca^2+/Mg^2+}$ is performed using the same pH 10 $\ce{NH3/NH4^+}$ buffer, with the addition of a small amount $\ce{MgSO4}$ and $\ce{EDTA}$ to the buffer.

As far as I know, and according to a fast online search, the addition of magnesium is to ensure the visibility of the end point when using Eriochrome Black T (EBT) as indicator. Supposedly, the color change wouldn't be easily spotted without the magnesium addition. This is how it's supposed to happen:

Considering there will be no change of color in EBT only due to Calcium, a small amount of magnesium is necessary to interact with the indicator and dye the solution. The equivalent number of moles of EDTA is added to the buffer, otherwise the results would be overestimated. Upon completion of the titration, the remaming $\ce{Mg^2+}$ complexed with TBT would be chelated by EDTA and the endpoint detected.

This is important in basically two scenarios:


There is no magnesium in your sample.
The buffer used is not pH 10, but pH 12, exactly to remove magnesium and detect only calcium.


In both cases, you need some (known) amount of magnesium to see the color change.

The problem with this is that Calcium is supposed to color TBT. It's even the example given on TBT's Wikipedia page, and there's also a full article about the interactions of magnesium and TBT. So why would you add magnesium?

To answer your queston - which also became my own - I did a qualitative experiment, because I'm in the lab right now. And the result was interesting.

I added small amounts of Calcium Chloride and Magnesium Sulphate to two different beakers containing purified water, and then added a few drops of TBT. I also added TBT to the purified water as a blank. This is what I got:



As you can see, the $\ce{Ca^2+}$ solution did change the color of the indicator, but not nearly as much as $\ce{Mg^2+}$ did. If you think about real-sample analysis, where usually the water (or soil extract) contains other coloured materials, making sure the endpoint is easily visible is very important. As a colorblind, I already find this change in color a bit hard, and for $\ce{Ca^2+}$ alone I suppose it would be even harder without the magnesium.

So, finally: The Magnesium is added because the color change is easier to spot, even though calcium changes the color of the solution in presence of TBT.