What cable do a need to connect a SATA SSD to a computer with eSATA and USB plugs

The 5v power brick needs to be at least 1 amp of output, otherwise it will require USB power also.

See this document, near the bottom in the troubleshooting section.

http://www.vantecusa.com/system/application/media/data_file/nst-260_user_manual4.pdf

.

The device works well with all of my USB A ports (it comes with a very thick USB A to A cable)

That's one classic issue with the type-A port used on a USB device. Type-A ports are designated for USB hosts. You were using illegal A-A cables to connect it. Fine.

Now you are using a Type-A to Type-C cable. This cable is designed to connect USB host (type-A) to Type-C device. Therefore it has HOST SIGNATURE on CC pins in the cable's overmold. Your new Type-C is also a host, so you are attempting a host-host connection, and this fails for obvious reason.

To get your SSD enclosure to work with Type-C host, you need a Type-C to Type-A Female adapter, aka "OTG adapter.

This adapter will have DEVICE signature on Type-C end, and FEMALE Type-A port, which will act just as an ordinary PC port. Then use your illegal A-A cable.

There is no single-piece Type-C cable which will connect your illegal device with Type-C port, unless you do a serious overmold "surgery" and switch the embedded pull-up to a 5.1k pull down on one of CC pins.