Several things to consider with your question. The biggest factor is how do you use your vehicle?

To safely cover every scenario, yes is the only answer. If your spouse and especially children are in it- sell your blood and put good tires under them!

Older tires have 2 big problems:
1. The rubber gets hard and significantly reduces traction, ESPECIALLY in the wet, which can be particularly dangerous if you aren't knowing of this risk and driving accordingly. (This can be fun if you like to have fun with the skinny pedal and know how to do it safely.)
2. Reduced tread depth can cause very hazardous hydroplaning on standing water, even seemingly smaller puddles.