It's completely normal for the 1st gen Apple Pencil to behave this way if the battery completely drains out. It needs a little more power to wake up, hence why charging via the iPad might not work. The best way to revive it is by using a power adapter, as you've been doing.
Regarding the USB-C to Apple Pencil Adapter you bought, this accessory is actually designed for the 2nd generation Apple Pencil, which uses a different charging mechanism (magnetic charging via a USB-C port on an iPad Pro). The 1st gen Apple Pencil is designed to charge with a lightning connector, either directly from the iPad or via the Lightning cable with the little adapter it originally came with. Therefore, the USB-C to Apple Pencil Adapter would not work with the 1st gen Apple Pencil, which might be why you're experiencing difficulties.
For the time being, you might need to purchase a new Lightning cable and power adapter to charge your 1st gen Apple Pencil, especially if your battery often dies completely. Another option could be to try and borrow one from someone else to see if that works.