Electronic metronome Wien oscillator
I think, in principle your analysis is correct. And - yes, the resistors R15 and R16 provide a DC reference which enables a dc operating point (output voltage) at app. half of the powwer supply. Therefore - in contrast to your assumption, the output will consist of a DC voltage and a superimposed full sine wave.

The frequency of the sine wave wo (provided the diodes allow a good amplitude control) will be - of course - in accordance with the Barkhausen criterion. In this context, you have nothing to do than to find the mid frequency wo (center frequency) of the RC-bandpass in the positive feddback path. At w=wo the phase shift of this bandpass will be zero and the feedback factor 1/3.

Comment: The above mentioned facor (1/3) applies only if the bandpass in the positive feedback path is symmetric (equal R and equal C in the series resp. the parallel RC combination). In your case, there is an additional 10k pot (R20 ?) in series with the 30k resistor. This resistor slightly detunes the bandpass and, thus, can vary the oscillator frequency within a certain tuning range. As a consequency, the feedback factor (nominal 1/3) will slighly change as well. This will be, however, not a problem as long as the amplitude regulation mechanism can cope for this variation.