I kinda thought it may be the PSU and was about to swap that out.
Are you still taking repair jobs on motherboards. I still have that 3000T board damaged and could ship it to you ? Probably not until the spring though.
It could be the power supply, or anything else, though you'll save yourself time with a few basic checks. For instance, if the power LED is on, the board probably has 5V, which is all that's needed to boot to the 'insert disk' screen.
The power supply ticking noise could be it going into over-current shut down. So a short to ground on any of the outputs may be causing it not to work. If you were to unplug all the loads, does the power supply operate normally? You can use a meter to check the 5V output is valid. Then gradually add the loads to the point it stops working to narrow down where the problem is.
Yes, I'm currently accepting Amiga boards for repair. A word of caution with the A3000T; if it's the usual case of leaked battery, these are notoriously expensive to repair for a number of reasons. Their large size and weight makes international shipping a bit pricey. And the area of battery corrosion in this model is particularly unfortunate as it often damages the FPU and the dozens of tracks in the area which route in and out of the 4-layer board, which usually breaks various ROM and real time clock connections. The physical layout of the tracks in the corrosion area (dense population with many vias routing to internal layers) can make it very time consuming and therefore expensive to repair, and the job can get ugly and become fragile if there are many tracks to repair. Don't mean to put you off, this is just my experience over 15 years of repairing them.
If you can send a detailed photo showing the areas of damage, I can estimate how practical it might be to repair.
http://amiga.serveftp.net