Can't get DS1077 programmable oscillator to work
I would recommend reading the datasheet again for that device. It sounds like CTRL0 and CTRL1 can also be used as output enables depending on the devices control registers MUX and BUS. This would explain why when you assert CTRL0/CTRL1 the oscillator output is disabled. Another thing to note is that the datasheet says the register values are store on an internal EEPROM, but only after a write command is executed. It could be possible that you think the registers are set one way, but when you power cycle the board the previous values are being loaded from EEPROM. I've worked a lot with I2C clock chips and every time things weren't working it was because I miss interpreted one of the register values. Write each register out on initialization then read them all back to make sure you initialized them properly. Compare the results to the datasheet to see if they make sense.

EDIT: It appears that the CTRL pins can also be used to place the device in a power-down mode which would explain why it's not responding to I2C at times.