Glad to be of assistance.
Looking at the schem the tone controls are seperate to the amp itself, audio going to pin 9 of the ic so personally I'd keep the board and make a little board up with an amp ic on it and hot glue it to the audio board. I notice there is what looks like some Eq on the existing amp, a new amp might sound a bit different.
If you can get the original ic it would make things a lot easier, might be worth looking into whether the upc101 is the same as the La4201, it wouldnt surprise me if they are interchangeable, the manufacts may have swapped out the chip from the schem as they got a load cheep.
Someone of your capability and interest ought to have a transistor tester, you can get them really cheap from ebay. You can test a tranny with a multimeter to an extent, theres loads of stuff on youtube about this, personally I use an octupus component tester, this does require a 'scope and a little practice.
Heres a super simple circuit for the Lm386, you can omit the cap between pin 1 and 8 to reduce the gain if the volume pot is too sudden, lower values like 4u7 2u2 etc give reducing amounts of gain. Looks like on your old amp chip pin1 is +supply and pin 12 is gnd, so long as you pull the original ic you could connect the o/p of the Lm386 amp to pin 14 and use the original speaker and 'phone jack, in fact all the wires to the Lm386 can go back to the original audio amp ic connections.
https://www.eleccircuit.com/lm386-audio-amplifier-circuit/