As described, your iPad battery may have reached the end of its usable life. If the battery has failed, the iPad will not successfully boot to a running state. When connected to its Power Adapter, the iPad may be able to boot to a running state and/or recovery mode.
However, when connected to your computer, the USB port may be unable to provide adequate power to the iPad - and therefore will not restart. Many computer USB ports cannot source sufficient power to charge an iPad. Unless explicitly designed for charging connected devices, computer USB ports are typically limited to 0.5-1A @5V (i.e., 2.5-5.0W). Most models of iPad require at least 12W (2.4A @5V).
Be aware that attempting to restore an iPad with a failing battery, from recovery mode, risks "bricking" the device.
Are you able to use the iPad while remaining connected to your Power Adapter? If yes, while monitoring the indicated battery state, disconnect the Power Adapter; if the indicated battery state suddenly falls, or the iPad shuts-off, this test demonstrates the iPad battery having reached the end of its usable life.
As you have an original first generation iPad, battery service is not available - and neither are compatible Apps still available from the Apple app Store. Replacement of the iPad with a much newer model is the only viable option.