My home is not small, but I cannot imagine a 70-200 being the ideal lens indoor, since it isn't a hangar, and the space is all broken up into rooms.
I tried it lately, it does work. But of course this kind of a lens is bought for other stuff, too. Next summer my boy is going to school, aka larger indoor rooms where plays and sports happen.
Have you considered a few good primes instead for indoors? Only you would know what focal length fits, but do remember you can crop if the framing is not perfect. Indoors it is unlikely you would be cropping more than 50% even in a worst-case scenario. A good 35 or 50 should do the trick I would imagine.
Sure, I am using a 50/1.4G that I bought used for a price in between a new 50/1.8G and 50/1.4G. I am contemplating getting a 85/1.8G, but it's likely better to put that money into a 70-200/2.8 instead.
I still own my Olympus E-M5 + 45/1.8, too. And even its 90/3.6 equivalent DoF can be too shallow to catch them moving and leaves little room to put more than one in a shot without stopping down further. Fortunately they keep still sometimes...
But then they often don't even when the feet are not moving. Then a bit of Photoshop's anti lens-blur sharpening might be needed to rescue a 1/200s shot.
Yes, the 24 mp sensor of the D750 allows for quite some cropping, too. Which also helps with stuff like minimum focus distance and allowing the kids some breathing room when you don't have a longer lens attached.