I thought maybe using the tuning capacitor,
but other connections to it would interfere with the radio.
Maybe there is a spare capacitor gang in the tuner.
Also thought the value off the tuner would be a bit small to use a resistor
to time the discharge of the tuning capacitor.

Yep, that's how they jump from station to station instantly, they just generate the frequency digitally, and already know what the station frequencies are. Some of the very cheap radios just have an up/down button pair, that jumps between stations.

I was sure it had a tuning gang that had to be turned, and wondered how the LCD worked at the time.
I'll sacrifice one and bring for show and tell next time. I should have looked harder.
That will take a week for pay day hobby budget though.
I got one of these a while back and wrecked it because it wasn't a proper PLL radio.
If you could change channels with buttons, and station memory, I'd have kept it.

This topic is quite interesting to discussion. I think, It will be happening with the internal frequency counter. Generally, Frequency is defined as the number of events of a particular sort occurring in a set period of time. The accuracy of a frequency counter is strongly dependent on the stability of its time base.

If the chip is a frequency counter then where is the clock coming from?
The old fashioned way?
The chip FM radio I made on protoboard with a radio in a chip,
has enough inductors and capacitors outside the chip for tuned frequencies.

I should go get one tomorrow, I think it's too dumb to jump between radio
stations. If it was able to, it would be made that way, so I'm inclined to believe it's analogue.
I can afford to wreck a cheap radio on the hobby budget.

You can tap off the signal from the local oscillator (LO) using a FET.
You need to know the intermediate frequency (IF) being used in the radio.
Then you have to add/subtract (depending of the radio design) the IF to/from the LO frequency.

If you are building your own frequency counter there is a way to add/subtract the IF.

Here is a photo of my DX radio with a homemade frequency readout showing the IF offset that is preset into the counter.
I can choose to add or subtract the IF offset.

Thanks MrChips, I will get onto you when I'm actually doing it.
The frequency counter isn't hard, it might be interfacing to the radio,
and/or buffering the clock without wrecking the clock signal.
I do know the Intermediate Frequency.

What has me a little puzzled since last posting is the frequency counter,
even the one you have pictured, if not this radio, must be run by a chip
significantly faster than 108MHz to be useful for FM,
and I don't think there's a micro that fast in a cheap radio to dodge putting a PLL in it.

Unless the higher frequency can be divided, at the cost of resolution,
or there is a relative, but lower frequency available somewhere else.
But that's not consistent with what you so far explained about a super-het.

The radio I linked has been discontinued by the retailer,
so I got this one instead:http://www.jaycar.com.au/productView.asp?ID=AR1736
The guy that sold it to me assured me it's more or less the same thing.
It has a belt drive in it to make the tuning knob more sensitive
because the tuning knob on the case can do 2.5 revolutions,
but it's still an old fashioned tuning gang.