Afraid I can’t offer any tricks, and unfortunately I will be doing this task myself soon. However one thing that concerns me about you seeing leaks is that if there is a full pan in the bottom, that pan should have a lip on it that goes up behind (and higher than) the edge of the tile. This means any water at the height of the caulk should not be causing a leak. Is it possible that the mess of caulking you are seeing is due to a previous owner attempting to cover over actual holes in the pan?
I had a plumber out to look at my basement shower right before Christmas. The drain was backing up and we couldn’t get it to clear, but even when running the water slowly (nothing appearing to come back up the drain) I was seeing water running out onto the floor after a couple minutes. Well it turns out there is supposed to be a seal between the drain pipe and the larger pipe that drops down from the pan. My seal was completely missing so the water was running onto the packed earth under my cement floor and then backing up underneath the shower pan. He told me I could seal this with silicone and this has been working just fine so far, but I’ve also found rubber seals I can buy to hammer into place for a more permanent seal next time it fails. Not sure if any of this helps with your situation, but I figure good knowledge should be passed around!
Fortunately I know where the leak was coming from. It is where the surround meets the glass front and pan. The surround is plastic, not tile, and was clearly leaking from the corner.
I did think of one thing I’ve seen at the hardware store… they make a tool with a triangular tip, which I believe was for removing caulking from corners. You might take a look for one of those?