If I have to be critical...
Why on Earth are the amplifiers fed from an unspecified 13.5V supply but the motor fed from a chain of three regulators. Doesn't make the slightest sense to me.
It is unusual for motors in cassette mechanisms to run from 1.42V but regardless of that, a simple single stage regulator, ideally a switching type will work perfectly well and will not introduce any audible noise. That would eliminate 3 ICs and hopefully kill the motor noise in one step.
DC erase is not a good solution. It works but the erase head will to some extent record noise from the supply. it is basically the same as the record/playback head but with a wider gap and hence no ability to impose high frequency fields to the tape. It will still record low frequencies though, and running it from the incoming supply might add to the recording noise instead of erasing it.
In record mode, there is no bias to the record/playback head. This WILL introduce distortion, primarily at the second harmonic of the audio. There will be hysteresis to overcome as the audio signal reverses polarity. It needs bias, ideally AC bias or the same problem as the erase head will occur.
That 100nF capacitor between the stages really should have a resistor in series with it. The second stage in particular will be very prone to instability without it. The gain is set by the ratio of feedback path impedance and the source impedance. As the source is the output of the first stage it will be very low so the second stage will be running at very high gain. The 100nF capacitor is essentially the only thing controlling the gain and given that it is reactive, the gain will be higher as its reactance drops, in other words at higher frequencies. This is exactly the opposite of what you are trying to achieve.
Brian.
[edit] I should point out that in a previous life I worked as design engineer for a well respected studio recording equipment manufacturer - on the design of tape recorders.