Delay Low Signal, Fast High Signal
The problem with your transistor circuit is that you have created an emitter follower for the device connected at your output. This means that there can never be a saturated base current turning the transistor on and, as soon as you the close the switch you are reducing the base voltage by discharging the capacitor and immediately you see this reflected on the emitter. This is why it's called an emitter follower.

Ideally you would use an emitter that is grounded and put the "device" in the collector and up to Vcc. I would aldo use a changeover contact to remove the capacitor charging resistor and replace it with a grounding resistor. In other words have the capacitor fed by a charging resistor or a discharging resistor but not both at the same time.