... I haven't had any instances where I've felt like the stand was unstable...
Ah, man...you've been lucky... so far. Mr Murphy is always present on brew day. So consider the following:
To create a severe burn (2nd or 3rd degree), a water temp of:
120 F takes 10 minutes to result in a severe burn;
125 F takes 2 minutes to result in a severe burn;
140 F takes just 6 seconds
to result in a severe burn;
150 F takes just 2 seconds
to result in a severe burn.
mashing is done around 150-158 F and sparge water is often hotter (around 170-190 F)--so you're talking about having the potential of 10+ gallons of scalding hot water...
Another possibility is to make a wider base (2x4's, angle iron, etc.) and mount it to your telco base to give more stability. The center of gravity (moment of inertia) is just too high for that narrow base you currently have. It would just take a bump, a nudge, a tug on a hose, or a stiff wind and some bad luck and someone's having a bad day.