Would the battery loosing a charge make the tach not register?
No, not necessarily. But a bad battery terminal connection can and will.
Your car / truck battery is pretty much maintenance free these days. We don't hear much about automotive battery maintenance anymore. Used to be when you did a "service job" on your car, you changed the oil, topped off fluids, including the battery, cleaned the terminal ends, and hit the 14 grease zerks on the chassis. Motorcycle batteries fall victim to neglect mostly because they are so well hidden....out of sight, out of mind.
Today, Harry Car Owner has a maintenance free battery, and an out of work accountant changes his oil at the Lucky Lube. Most motorcycle batteries have not followed the automotive industry into the current century. Unless Joe Motorcycle Owner is satisfied with getting one riding season out of his bike battery, it may require a little maintenance effort on his part. Or the rider can trust the assessments of the commission motivated staff person at the shop they use for everything more technical than adding gasoline. These staff people will send you Christmas cards from the Caribbean out of gratitude. The more "professional" staff people do not include lines in their greeting that mention that their trip was made possible by your ignorance.
Another thing about motorcycle batteries.....
An effective battery maintenance schedule begins before
that day you hear that ride spoiling "click" when you hit the starter button, or more disturbing, nothing at all. To maintain a conventional motorcycle battery properly, check the levels of water and battery acid and top them off. Always use distilled to de-ionized water to avoid contamination. Periodic cleaning of terminals not only insures that power can flow to the starter, it will allow power to flow back in to the battery from the alternator. Even a maintenance free battery will have it's life span extended by periodic charging. Battery tender type battery maintainers are not frivolous investments.
In this case, this battery in Mike's KLR needs to be checked for charge or bad connection if he wants to salvage his battery. A battery which is fully discharged and left uncharged will sulphate and will not be able to retain or sustain a full charge for any length of time, even if you have been able to fully charge it. Allowing it to set uncharged is just signing it's death certificate.
Though rare, batteries can freeze and crack their cases. Little is sadder and more traumatic than watching a KLR rider reach into their pocket to pay for a new drive chain and swing arm because their's was eaten by battery acid from an unmaintained battery. The acid that drips from the swing arm and chain onto the concrete floor makes for an interesting non-skid surface, if there is a positive note to be found in this.
One other suggestion regarding winter storage I would offer would be to add some fuel stabilizer such as STA-BIL, then fill the bike's fuel tank. Turn off the petcock and run your bike until it quits. In most of North America, the gasolines available to us are a minimum E10 ethanol blend, which can go sour in as little as 10 days and gum up your carburetor. Otherwise, there will be threads available and very active come spring regarding carburetor disassembly and removal of carburetor jets and the various cleaning methods of orifices and passageways.