So as you say, the problem is almost certainly isolated to the turntable. A meter to check continuity would speed your troubleshooting up considerably. As I mentioned to someone in another thread recently, you can get a decent one for about $12 on Amazon US, if that's where you are. It practically pays for itself the first time you use it.
The wiring job looks just a bit too haphazard compared to, well, other pictures of this turntable that I've seen so far. Are the tonearm wires that are soldered to the terminal actually insulated? If not, then they will short out if they come into contact with each other or the other pole on the terminal. Or the insulation at some section on a pair of wires may have been damaged if you caught it with the soldering iron, which for wires that thin will melt through in an instant. If you got the pos. and neg. wire at the same moment, that would create a possible shorting point. Also, though it looks okay in the picture, make sure that you haven't accidentally bridged the positive and negative terminal of the RCA with solder. Again, easier to check with a multimeter.
Looking at the pictures in the other thread, one thing that makes it less surprising that this happened in the left channel is that the negative lug has had a lot of solder slopped onto it to connect both the ground and the tonearm wire. It doesn't look like a great joint. Don't worry, soldering quickly improves with practice - but fwiw, thick speaker wire is not what you want to be soldering in the first place. The wire is probably not suitable for your turntable, and your soldering iron is almost certainly not suitable for that gauge of wire.
If you want to check continuity for the left channel (you do), put one probe of the meter onto the appropriate headshell contact pin in the tonearm, and the other probe onto the negative lug (not the tonearm wire) down this end. If there is no continuity, you will have to redo that connection, because it would indicate that the tonearm wire is not making contact with the RCA. Best way is to start again on that lug, just remove all the old solder and try again. The practice is good for you. I prefer braid for desoldering. Not a fan of the hand pump method, because I'm rubbish at it.