Peter,
It is up to you if your application needs to be energy-efficient and therefore need EN2 (sleep) wired to MCU.
From the datasheet: "When first establishing communications with the TRF7970A, the SOFT_INIT (0x03) command should be sent first from the MCU". I wanted to check if you are sending it but your reply is not clear for me. Please ensure that software init is a first command you sent to radio.
What I do and works for me: perform reset on EN1, check default register settings, send soft init, send IDLE.
Radio is not the easiest device to use I found timings critical for its operation. Be sure that you follow TI recommendations by checking all timings with logic analyser and compare them with TI values (there are analyser dumps available on the forums for TRF7960).
What is your SPI clock? I would recommend 8MHz (I use parallel only in my application).
Be sure that you are familiar with errata a.k.a. "Firmware Design Hints": http://www.ti.com/lit/an/sloa159/sloa159.pdf
Regards,
Maciej