You'll want to pop the board off the case by cutting the heads off the rivets, and entirely remove the display module

NO NO NO, A thousand times NO. Do NOT do this as the first order of business, the calculator will never be the same, keyboard wise. While a battery leak can cause an open trace in the RC tank circuit, there is zero need to forever damage the keyboard action as the open trace can be repaired from the component size. Popping heat stakes is the absolute LAST resort. BTW, the symptoms you describe do sound like a dead clock which would be the result if the tank is open.
Clean up the obvious damage and see if that fixes it. If not, check the pc board photos over on the HP Calculator Museum and you can see where the LC parts are under the battery box area and follow the traces. I can post the pinout of the nut processor which gives you the RC tank connection points if needed. They might posted over on Eric Smiths site, brouhahacom. I do have loose boards and can provide the connection points if needed.
Last thought. Yes, solder joints do fracture but usually in the center area of the board where the most flexing movement occurs. Fractures can easily seen with 5X or more magnification. You don’t want to be spreading flux and reflowing the chips, you only want to repair any visibly broken connections.
Edit: the nut processor is the far right square chip. Part number varies. The RC tank lands on pins 7 & 8, note that pin 6 position is blank/missing but counted. If you check with an ohmmeter between 7/8, you should see about 6-7 ohms if the traces are intact. Pins 7 & 8 are on the bottom of the chip, the side closest to the keyboard. If you need help this, let us know and we can markup a photo and post.