The adapter in the link violates the USB specifications and is a fire hazard. You should bust it to pieces with a hammer and throw the crushed bits away. Seriously. This adapter runs the risk of short circuiting some very valuable computer equipment. It's a $7 mistake waiting to become a $700 mistake.
I recently purchased a Satechi USB-C to Gigabit Ethernet adapter for the USB-C Macbook and would like to use that also for the older macbook.
You can do that with the right adapter but it would be cheaper to just buy the correct style Ethernet adapter for the older MacBook.
I purchased a USB-C to USB-A converter and tried to connect the USB-C adapter, but it does not work.
That's probably because the adapter was made broken. This adapter violates the USB specifications, it should not exist because the specifications specifically say this adapter should not exist. Throw it away before you plug in the wrong cable to it and damage something valuable.
Here's a Thunderbolt to gigabit Ethernet adapter to consider.
Connecting USB-C devices to a computer without a USB-C port is not trivial, it cannot be done with a $7 adapter. It would take a PCIe breakout box and USB-C adapter card, which would easily cost more than just buying the right gigabit Ethernet adapter for your older Macbook.
Rereading my post I can see that I might come off as being mean. I'm not trying to be mean, I'm trying to be stern and straightforward. This is a stern warning that the USB-C to USB-A adapter you bought can do expensive damage to your computer, it needs to go away before that damage is done.