Would you trust 13 year old tires?

short answer, feck no.

Some have mentioned tires rot from the inside out, but thats not the case. What breaks a tire down is exposure to air, heat, and UV.
Your tires may appear new, but the compound has degraded and they are no longer structurally sound. This means if you are lucky, they will develop a slow leak to tip you off, but most likely the failure will occur at speed, and be sudden and catastrophic as the sidewall fails. This can result in a damaged rim, damaged sheet metal, and total loss of vehicle control. So you are further tempting fate in a vehicle that isn't exactly the safest thing on the road to start with. It really doesn't matter the mileage on the tires or the looks. Over time they break down, end of story. Ever went to pull an old blue tarp off something, tarp looks fine, but once you give it a yank, it disentegrates in your hand. So yea, you are cruising around on that old blue tarp.

I'd also argue the multiple comments about tires from back in the day being made better and lasting longer. Tire technology has come a very very long way.
You can easily get a quality all-weather tire for under $90. It's not if, but when you come up short on tire roulette..