*Could you briefly explain 'IC's' to me, please? It sounds like a very precise soldering job you've mentioned? I do have a couple of old donor boards I'm happy to practice on to see if my skills are up to the job.
IC's are monolithic semiconductors that typically looks like flat rectangular shaped centipedes. The acronym is for Integrated Circuits and a simple description is that they are tiny circuit boards populated with electronic components and circuitry, then encased in an epoxy shell with leads protruding for wiring them up. In addition to far lower cost than making up the same circuits individually, they are packaged this way to allow the repetitive use of the same circuits in different applications. In any event, after your updated description of when and how the stereo light comes on, it's pretty clear that the IC's are "probably" OK and that some adjustment is what you need. There are 2 possible adjustment scenarios.
#1, the stereo pilot signal adjustment is off and this is something that you can probably do yourself and is relatively safe in that misadjustment is not catastrophic and relatively easy to undo if you mess up. In this case, an onboard potentiometer needs to be rotated to the centerpoint of the range where the stereo light illuminates. If you have a service manual for this model, then you can leasily find which is the mpx stereo decoder and the associated stereo adjustment pot. Lacking the service manual, there is another way to find which pot this is, it's generally easiest to start by looking up the datasheets on the various IC's in your tuner board. Datasheets will usually tell you the purpose of that IC, such as AM/FM IF amplifier, MPX decoder, etc. Once you've figured out which IC is the MPX decoder, then the datasheet will usually provide a sample circuit. Luckily, you will find that most electronics manufacturers, with few exceptions, generally follow the sampe circuits very closely. In this case, search for the variable resistor in the circuit that corresponds to stereo, 19khz, pilot, etc. That is the pot to adjust. Do not adjust the one that is indicated for separation. It sounds hard, but it really isn't.
#2, IF frequency or Local Oscillators are out of adjustment or alignment. These aren't really DIY type of fixes due to that you will require a certain amount of rather expensive instruments to perform these, as well as a knowledgeable technician. Some folks have tried experimenting by messing with these adjustment points (and there are many when you get to this point and some require bending of the coil loops to accomplish) but I don't recommend it. IF we are speaking of AM, you might luck out since many AM adjustments simply require peaking and you can sorta of hear the difference but the FM adjustment is far more complicated and it's possible, likely even to throw the alignment out further, possilbly rendering the FM section almost useless.
#3, Band coverage. This adjustment is performed to ensure that the range (i.e. 88-108) is properly adjusted. When off, this is condition is typically evidenced by the stations not lining up to the proper spot on the dial. This adjustment is possible for the DIYer to perform if you have accesss to a FM signal generator. The reason you need the FM signal generator is that you need the generator to create the FM signal for the adjustment. Typically, you will have the generator output a signal at the specified frequency indicated in the service manual, usually a range extreme such as 88mhz, tune the tuner-dial to the same frequency then adjust at the specified locations until the tuner receives the tone outputted by the signal generator, then do the same for the other extreme such as 108mhz etc. and this procedure is repeated back and forth and repeated until no further improvements are evident. The reason is that each adjustment not only affects the reception at the current point but also has affect at other end of tuner dial too. By the way, an FM signal generator is not the same as an audio generator since it needs to output in the RF range.
This all probably sounds overwhelming and it's a science that is not easy to understand, and almost impossible to perform properly without access to proper instruments. My only recommendation to you is try to see if you can find the MPX decoder chip, perform the adjustment and see if that helps. Or take hi-res picture of the tuner circuit board and we will try to find that chip and the adjustment pot for you. For some reason, I couldn't find a service manual for the RX-CS700, but do have one for the RX-CS710 amd 780. Perhaps it's the same or different, not sure.