my 2 cents worth - I think it should work this way, based on a similar style Melico unit I own and use, dating to the early 60's for it's design:
Your unit has two modes- one is a straight timer (Time), and the second is an analyser function (Balance).
Time operation shoud be straight formward - set the seconds dial for the time you want, hit the expose button, and the safelight power goes off, and the the enlarger button goes off for the amount of time set on the dial. Note that the time dial has a logarithmic scale; this is what gives me the clue to the next functional mode.
Balance mode puts the thing into an analyser mode. I presume it has a photo sensor - the thin wire coming out of the unit I suspect might be related to this function (The photo sensor you normally set under the enlarged image projected on th base board, usually on the densest part of the negative that you still want to preserve highlight detail in.)
You make a print for calibration purposes, using the fixed time mode dial, and note where there is a part of the print that shows the slightest detial in the highlights. Dont change the time, enlarger head head height, aperture or type of paper being used. Also develop the prints for a consistent fixed time, agitation, etc.
Put the unit into balance mode, and put the sensor under the projected image where the slightest highlight detail falls. Don't have any filters in the path when doing this calibration, and subsequnet analysis. Insert filters only after the bridge has been balanced.
Turn the paper speed dial until the thing is balanced - this may be when one or both of the lights go out; one might be for over balance, and the other for under balance, or perhaps one light tells you that the thing is turned on and the other light will come on or go off at some point when you rotate the paper speed dial.
My Melico uses a fine tuning indicator tube of the sort found in old tube radios.
Once the indication of whatever form is found, the unit has 'balanced' - likely a balanced wheatsone bridge for the electrical engineers out there.
Then, note the setting of the paper speed dial, and don't move it for as long as you are using this type of paper and developer combination.
Then while in balance mode, on the next negative, or the same negative at a different head height, you put the probe under the area where you want highlight detail, and turn the time dial unti the balance indication happens. Press the expose button, and the right amount of eposure will be given to get the detial in the highlight to the same tone as the test print.
The negative, if different, may be needing a different contrast range. There can be different effective paper speeds for fixed VC filters, so if the need is past say #3 in Ilford, you may need to double the time after balancing if you initially calibrated on aneg that printed well on #2. If using a dichroic head, I would urge you to test per 'the variable contrast printing manual' by anschel to get the filtration at constant spped settings figured out for your particular dichroic head.
There are also more advance ways to use the analyser to figure out the contrast range of the negative (ie when the probe balances on dark areas as well as highlights, and how far apart in time that they are (fixed paper speed) or how far apart in paer speed (fixed time)) to guide you to a good filter for the first test exposure.
There is a good chance that your unit, if mis-bahaving may just be suffering from dried out electrolyic capacitors. They typically poop out from old age after twenty or more years. These can be replaced if you are electronics handy at reasonably low cosy usually. I recently replaced all of the caps in the power supply part of a favourite old stereo receiver for $8, rather than figure out which ones were causing the supply to take a long time to stabilise and power up after turn on.
Hope this adds some clarity to your situation.
Best of luck if you do contact him!
