You could disconnect C1 from R1, then use a changeover switch with the common to C1, normally closed to +12V (preferably with a low value resistor is series, eg. 100 ohms) and normally open to R1.
That means the cap is always charged ready to start the circuit, but it only connects when someone stands on the platform and the switch clicks across. The circuit will then start cycling and slow down as C1 discharges.
That's the simplest way, the only downside is the "spin" would stop instantly if they got off the platform too soon.
I'd use something like a generic V3 microswitch with a lever actuator as the changeover switch.
eg.
https://www.ebay.co.uk/itm/Microswi...h=item3d66e8b79b:g:OycAAOSwMZFbl8Me:rk:8:pf:0