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 Does anyone know whether push-to-talk with the first mobile phones needed a cell tower?

https://en.m.wikipedia.org/wiki/PoC_radio 
 Yep

Push-to-Talk over Cellular (PoC) provides two-way radio services over 3G, 4G, and Long Term Evolution (LTE) technology, creating a nationwide radio network that utilizes the cellular infrastructure of Mobile Network Operators. This enables radio networks with very wide coverage area and untethers radio users from the range of repeaters and base stations used in traditional radio networks.  
 Yes.