Plus 1 on the above posts. You can't just add a throttle lever W/O other add on parts.
Your set up is very simple, works and really no reason to change it...
*If your the only one using the mower, a suggestion, for a simple "fix / mod".
On my Snapper, self propelled walk behind mower, I removed the safety bar and entire brake assembly-pitched it...
Then, for the coil wire, hooked up to a simple metal, on/off switch and attached to a bracket I made up (attached to where the brake was bolted).
So now, flip the switch on, engine runs, no levers-nothing. That is until you flip the switch OFF.
No re-starting, no handle to hold, much simpler, BUT the safety system is gone (have to remember to keep hands and feet from under a running mower-duh...).