Can Apple TV audio output be adjusted to work with old analog amplifier

No definitely it will not work ( sorry that is my opinion) there is no setting to set the audio signal analog for the ATV. But are some bright minds on this website , and sometimes give you unexpected solutions.
Good luck

However, when I tried to use the analog (white/red) audio outputs, connected Apple TV to my 30 year old Kenwood integrated amp, the sound was extremely distorted and LOUD.

What input type/level are you using. Many old amps/pre-amps have specific device load/level inputs. It may be possible to use special level/load matching cables or an impedance/level matching box between the TV output and amp/preamp inputs.

Is there any adjustment that can be made to the analog output from apple TV? or do I now have an excuse to go out and buy new audio equipment?

Some very, very old (usually "tube" based) amps/preams may have input level adjustments. I personally would probably use it as an excuse to shop for a new digital AV amp/receiver.

Not plugging into a phono input for turntables by any chance?

If so try tape/tuner/aux etc.

AC

The phono input is intended for very small voltages produced by moving magnet/coil cartridges which will be a couple of mV at most. This input also performs RIAA equalisation due to the way sound is encoded on vinyl which the other inputs don't need.

Line level inputs (the others esentially) accept far higher voltages of 100-150mV or more - connect a turntable to one of these and it'll be very quiet, connect a line level output from CD/tape/tuner/AppleTv to phono and it'll overload the circuit and sound horribly distorted.

AC