does Tera Term handle software flow control internally?
There is no standard "command" one side can send to enable or disable flow control on the remote side. Hence, your terminal will not send anything when you enable or disable handling of (software) flow control.

What you do enable is the handling of the XON and XOFF bytes by the terminal. This means that the terminal program will be aware that XON (0x11) and XOFF (0x13) when received are not to be shown in the terminal window but mean to resume or pause, respectively, transmitting data to the other side.

Or, in other words, XON/XOFF is only sent by either side when that side is currently receiving data.

Edit: The misunderstanding may be in that "XON" does not "turn on" flow control, but is itself a flow control command signalling to the other side to turn "transmission on", i.e. go on sending data; "XOFF" is sent to the other side to tell it to turn "transmission off", i.e. to pause sending.