Why doesn't my doorbell have power at the button?
It sounds like you've connected the two transformer terminals directly to the two chime terminals. That would cause the chime to sound immediately when the power is turned on, and no power would go to the switch.
The correct wiring looks like a loop or ring connecting the transformer, chime, and switch. See removeandreplace.com.

With your specific cables it could go like this:

At transformer, white wire to one terminal and black wire to the other terminal.
At chime, white wire from transformer goes to chime's transformer terminal. White wire from button goes to chime's front door terminal. Black wire from transformer connects to black wire from from button. Note: black wires do not connect to chime at all.
At button, white wire goes to one terminal of SimpliSafe and black wire goes to the other terminal of SimpliSafe.

Do steps 1 and 2 first, leaving the button wires by the front door disconnected. Turn on power. Touch the bare ends of the button wires to each other. The chime should sound.
After confirming the chime works correctly by touching the wires, turn power off again and do step 3. Then verify the complete system.