Gas bags
Yes, if gas turbines are left long term then yes, the lack of fresh oil supply will result in the bearings eventually running dry but you would have to leave them windmilling like this for weeks on end. Straps are provided to assist in safe working on the engines, boroscoping and other such activities. Some people will strap down when the wind picks up but actually the faster the shaft spins the better it will float and balance on the residual oil, but you wouldnt be able to safely stop it spinning to inspect the engine.
However, in a gas turbine that is used or at least run every so often then when it runs the oil in the bearings is refreshed and when it is then shutdown and left to windmill there is residual oil in the bearings and oil seals which prevents damage.
Like I said, by rotating it keeps the oil spread round and cushions the boundary between the shaft and the bearing and prevents 'egging' of the bearing from the weight of the shaft resting on the same spot on the bearing/seal by spreading the load and reducing pressure.
My source for this - I used to work for a major gas turbine manufacturer in their Whole Engine Modelling department, my specialty being in calculating rotor tip clearances, seal rub/wear profiles and movement/vibration patterns. And we would connect an N1 motor when shipping long distance to rotate the N1 shaft to ensure it arrived with the customer in best condition.