When I use AD7671 to convert signal (1Hz to 1kHz), I found steps in the output.
Attachment figure 1 is the waveform, where green line is input of AD7671;yellow line is the output of AD7671 (a DCA output, the DAC operated well).
I have checked the specs for the recommended ADR421 and compare with ADR01 in terms of the current drive capability and they are not very far. As you suggested about the driver. The AD8031 settling time is 125ns (at 0.1%) against the AD797 800ns at 0.0015%. This means that the AD8031 can settle faster but although the AD797 is slower but has higher accuracy when it settle. Using the AD8031 as driver, as recommended since this was proven to work with AD7671.