How does the thrust to weight ratio work in terms of aeronautics and rocketry?

I have been reading about fighter jets recently, and came across the thrust/weight ratio. Using the F-22 Raptor as an example, it has two engines that put out 104kN of thrust (without afterburners). This gives a combined thrust of 208kN. Its unloaded mass is 19,700kg, which is equivalent to a weight of 193kN. This gives it a thrust to weight ratio of 1.078:1.

All this makes sense to me. Where I have trouble though, is the implications of this figure on climb rates. Modern fighter jets can famously accelerate perfectly vertically, with some reaching Mach 1 within a short period of time. However, given a thrust/weight ratio of 1.08:1, it seems that the thrust of the engines is only just greater than the weight of the plane. The pilot should experience 1.08g of acceleration, and hence climb at 0.78m/s/s. This is obviously not the case, as evidenced by any number of aerobatics videos.

So where have I gone wrong? How does the thrust/weight ratio convert into climb rates?