I'm a litle confused about the complaints. But from what I can make out, one channel is out but tweeter on that side is working. Then there is question about a capacitor. Then there is a hum although it's not made clear when that happens. Then there is the disconnected black wire. Then there is complaint about what looks like resin. Then there is some complaint bout Lasonic not being responsive. Then something about paying a certified tech in audio who charged $118 to replace a resistor and some solder connections.
(1) If the tweeter is working but you hear no sound from the woofer, then it sounds to me like the woofer is blown.
(2) Like others have said, the black wire with ring terminal is to ground the deck. Without a proper ground, the deck can "hum."
(3) If the ground does not fix the hum, then the issue can be (a) broken head ground wires (b) bad electrolytic capacitors or (c) failing amplifier module.
(4) I see nothing wrong with that capacitor being there. Don't waste another minute of your (or our) life thinking about it anymore.
(5) the soldering process requires flux to clean metal during the soldering process in order to make good connection. Some flux requires removal after soldering to prevent corrossion. Rosin is very frequently used as a flux for use in electronics. It is deemed safe for use and non-corrossive. Many manufacturers leave it after the soldering process. Good practice is to remove it which you can do if it "bothers" you with a q-tip and isopropyl alcohol. Also, some contact cement is used to affix heavy components such as large capacitors. If not affixed, over time, heavy components can tear the solder traces from the boards. Lastly, those thick gobs on the tuner board are wax holding the position and shape of components in place to set them so the board can be aligned. Any movement of components, or even the shape of the copper wire coils will affect tuner alignment. The was prevents the movement. If you mess with those, don't be suprised if your tuner is all out of wack afterwards. In other words, leave it alone and move along... nothing to see here.
(6) I am puzzled as to why you think Lasonic needs to comment on a 35 year old item that is probably 34 years out of warranty. The boombox is deemed to be obsolete and no longer supported. Direct your anger elsewhere.......
(7)....... your "certified" tech.... I wonder why for what he charged you, that he hasn't determined the root of your problems. First, if the problem turns out to be a blown woofer, your tech owes you a refund of the entire service because that is something that he should've been able to figure out easily and if he missed it, then he isn't competent. However, there is certainly a possibility that the issue is in the circuitry and no Lasonic service manuals exist that I'm aware of which certainly makes troubleshooting circuitry 10 times harder. However, he should not have attempted the fix if he could not locate the manual since it seems he was not interested in investing the extra time necessary to troubleshoot without a manual. But then again, we don't know details, only what you mentioned in your post so I'll not blame him for now.... just that something doesn't add up. Hard to know what he charged $118 for if he couldn't even tell you what is wrong with your box.
I have a Lasonic 918 I got a a swapmeet for a $7.00 buy No B.S 
Powers up Radio/cassette works. Problem was Right ch full power. L /ch out. heavy Hum. turn bal all the way can hear radio/tape from Left tweeter Only 
Took to a certified tech in vintage audio. Bad deal. 
$118.00 basic resistor and some soldered connections restored. Tech found wires going to no where 
and grounds to??? sTILL BANGS but guys at Lasonic are straight up Crooked and wacked! 
guess should part it as a loss. pic of it is in my post( toys in a corner.)
it grounds deck for sure 
the pic of the box running is the tape, note LEDs are Peeking L ch is just showing ground hum sound right is responding to a jam called thanks to you/Cinnimon 
