Before I go any further, I must call a point of order...I cannot help but to detect a hint of passive-aggressive, sarcastic scoffing in the tone of the above post. I am unsure as to whether you meant it to be condescending in nature or not, but it certainly does have a rather derisory air which really is unneccessary and quite unwelcome.
This forum exists as a place for enthusiasts to discuss their common hobby and support one another, and there are many people here more than willing to impart their knowledge and/or experience and to provide possible solutions when help is sought.
The nature of the beast is that there are any number of possible causes for any given failure or ailment that may afflict a given machine... whilst these may not be simple aging and wear of electronic components, in m any cases these are a direct cause, or part of the state of dilapidation which is as a whole preventing the machine from functioning. There is EVERY chance that something else is still wrong, but it does not mean that those who suggested you start at basics were wrong or should be denounced because their initial suggestions alone did not solve a problem which is further reaching than simple out of spec caps or electrolyte shorts across connections.
There is also every possibility that these issues caused other damage, or that multiple faults exist in the system. The older the machine gets, the higher the likelihood of multiple concurrent faults developing in different systems and subsystems within that machine becomes. My 7600 for instance had a faulty RAM module, a faulty hard drive, and a power supply that blew up in my face, all of which had to be rectified before it would successfully boot... even now, it has a graphics artifact that requires further investigation to identify and correct what is obviously another fault in one of the machine's systems. The IIci predates it by a good 5-6 years so imagine how many things things could be wrong with it due simply to age and degradation.
Anyway, I'm not here to pick fights or get upon a moral pedestal, I am here because of old Macs. Furthermore I am not going to sit and hold a grudge based upon a disagreement, or get into a heated dispute over whom is right or wrong, or engage in a shitslinging contest, sohaving called my point of order, at this point I shall return to the subject matter at hand... a badly behaved IIci.
The battery should not make a huge deal of difference to the power-on sequence of a IIci, but if you feel that it could be the grail you seek in the quest for IIci goodness, by all means it will do no harm to replace it, as brand new units are cheaply and readily available on fleabay for very little money... My adivce would be to go with the Saft or Varta brand as a known reasonable brand. There are other far cheaper brands from Asian distributors and manufacturers that might seem tempting due to their miniscule pricetags, but personally, I wouldnt be one to trust them.
From what you are describing, it almost sounds as though the power supply has failed... I have had a number of different Mac's behave similarly shortly before failing to power on at all or having the PSU go bang. What portion of the unit has failed if this is the cause is entirely anybodies guess... the obvious course of action before anything, if you are 150% confident that all components on your logic board are servicable and that your solder work is sound, would be to open the PSU and see what you can see... if there are any obviously burned or blown out components, this is likely where your problems lie. Even if not, PSU caps fail just like any other electrolytic caps, so it would not hurt to replace them all as well, because like the ones you just replaced on the logic board, they are quite likely already failing and falling out of spec even if they don't yet appear physically damaged.
Ideally, to identify definitively whether you are dealing with further logic board problems or a power issue , you would be best served to get hold of a known working power supply and see if the machine functions differently... If not, you could be looking at electrolyte contamination, electrolytic rot of chips, or any other number of component failures on the logic board, or even the possibility of a bad joint on your new soldered components or even stray slag causing a short... Now I know one might be tempted to take offence to the mere suggestion that one's solder work is anything but exemplarary, but the reality of the situation is, solder is difficult to negotiate sometimes, and we are all human. None of us are perfect, and I have even seen highly skilled technicians pull their hair out for hours over faults that turned out to be a simple case of stray solder causing a short in a place that wasnt at first obvious. It could even be a broken or corroded circuit trace... just one is all it takes, and it's not hard to sever once with an accidental slip of a screwdriver.
