How did you power your old car radio?

Simple 12volt power supply brick, this one was a 1.5 amp one from an electronic organ that I picked up in a sale.

how would I go about stepping the speaker-level outs down to RCA levels?

Mine was a Blaupunkt from the 70's, with a line in/out on din plugs to attach a cassette recorder/player (They fitted them in Rolls Royce's at the time) - but as it's you, I went to all the trouble of hooking it all up again and got a perfectly good signal from the speaker outputs. I turned the volume down and very, very gradually brought it up whilst watching the record indicator on the cassette deck, but you're probably right, an attenuator would be a better solution.