The manual only describes the connection for the 5670, but you can use whatever tube and socket you like with the kit, as long as you can read the manuals and figure out which wire goes to which socket pin - and provide the socket if it is not a 9-pin. But to use an adapter to cover two different tubes with different pinouts and/or sockets, you must match them correctly.
The original Paramount had 12AT7 driver tubes.
The ECC88 has a different pinout.
The 6SN7 has a different socket (octal) as well as a different pinout.
The "sort-start" kit converts to 5670 tubes, which have yet another different pinout. The manual assumes you will replace the existing socket, and contains instructions for wiring with the 5670 pinout. To use a tube with a different pinout, or a different socket, you will have to identify the differences, and may have to lengthen some of the wires slightly.
You can install the socket of your choice, and wire for the tube of your choice (within reason!) - but you'll have to figure out the wiring yourself unless you use the supplied 9-pin socket and 5670 tube.
The 5670 pinout is not common, so it may be difficult to find an adapter if you wire for the 5670 and use an adapter to try the 6SN7. In a short google search, I found one adapter for using a 5670 with a 6SN7 socket - you could wire the "sort-start" kit for a 6SN7 and use such an adapter to use the 5670.
