Yes, I think the LCD needs a "high" voltage, meaning higher than the input voltage, but probably in the tens of volts, not hundreds. So maybe the board has one or more voltage doubler stages.
I can't confidently say you are wrong but I think you are wrong. LCDs don't need high voltages. You could settle this by checking around the circuit with a DVM. There are people on here who know this stuff better than me and I invite one of them to comment.
I found that touching the pins of that small 8 pin chip at the top right stopped the high pitch whine for a moment. There was a 50~100Hz buzz and the display went blank, until I removed my finger.
That makes me suspect a faulty chip or a dry joint or a broken track, I suggest you check for any of these. If touching the chip silences it then maybe, just maybe, some glue will do the job. My comment about wax was only half a joke, wax has the advantage of being easy to remove.
LCD display? Well, is a bit like Mount Fuji or River Avon
Or my (least) favourite 'personal identification PIN number' :o