Something that hasn't been mentioned, and 2 common issues that can cause an A600 to have a really long black screen before doing anything..

1, what version of Kickstart rom is fitted? If it is a 3.1 rom these can sometimes take ages to initialise. The A600 would have originally had a 2.0 rom installed.

2, does it have an internal hdd fitted? Some old hdd's can have a really long spin up cycle that takes longer than the Amiga takes to do its initial bootup cycle and so the Amiga gets stuck waiting for it and needs to be soft reset.

3, does it have a 1MB ram board fitted in the trapdoor expansion? These can go bad due to onboard batteries and/or fitted somms going bad. If there is one remove it and see if that helps.

Which brings me to a final question. When you do eventually get the Amiga to work, what does it do when you soft reset it? (Ctrl+Amiga+Amiga).

Just to mention. If you do replace the capacitor, when you have removed the old one, if it looks like it has leaked remember to neutralise the leak and then clean it up before you replace it with a new one. Lemon juice should be good for this, followed by isopropyl alcohol to clean it after.