Yep, the battery is toast. When you put the new battery in check the charging system with a meter. At idle it will show a very low charge from the machine, but as you raise the throttle so should the charge voltage. If it is not charging, let us know because there is still a wiring connection or bad ignition switch issue happening.
Something that also helps:
A few years ago, I found out that if I attach a solar panel charger on my battery it keeps the electrolyte active at a very low charge rate. By doing this it prevents battery sulfation which is what kills battery cells when the battery is left sitting for long periods.
I started this "battery life experiment" with my large Generac 15KW standby/ portable generator and that battery is now 4 years old.... Believe me, No One wants a dead battery on a generator when a power outage occurs. I run my generator once a month for an hour and the battery shows no signs of losing cranking amperage since I added the solar charger.
Since then, I have added a solar panel maintainer for use with my rider mowers and other equipment that require batteries. It has saved me a lot of money on annual or seasonal battery replacements. Most of my equipment is outdoors year-round, so sunlight is not a problem.
For my motorcycle, I use a 2A battery tender during the winter months when it is stored inside.