Why doesn't my doorbell have power at the button?

Your wiring scheme at the chime is less than clear from your wording in the comments, and your pictures don't show it. Sounds like you have not connected the black from the door or the white from the transformer to anything - or if you have, you didn't say to what.

Honestly to start with, since you do HAVE a multimeter, you should check the front door wire - first with both ends disconnected to make sure that there isn't a short (should have no continuity / infinite ohms between the wires) and then with the wires at the end you are not testing from twisted together (should have continuity / low ohms.)

That verifies that the wire is intact, and works properly, and the two ends you have are actually the two ends of the same wire...

Wiring at the chime location would depend on the wiring diagram of the system you are using. Most have connections on the chime (or in its box) for all the wires - i.e. two for the transformer, two for the front door, two for the back door if any, and needed connections between them are "built into the box."

For some "less advanced in the art of design" products you would need to connect one wire from the transformer to one wire of the doorbell directly, and the other wire from each would connect to the chime.

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:

1. At transformer, white wire to one terminal and black wire to the other terminal.
2. 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.
3. 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.