A box cutter or razor knife will work well and quickly to cut out old caulk, and will reduce the risk of unnecessary damage that might occur with a power tool.

Sometimes, there are no short cut for doing things, and saving time. If what you are doing, works well, then stick with it. Multi tools are useful for some things, but, probably not much for general maintenance.

I can't vouch for caulk removal but you will never regret buying an oscillating power tool. They make so many near impossible cutting tasks possible. They are pretty easy to control so damage, if any, should be minor and, I assume in an area that is going to get re-caulked anyway.

The blades are really expensive at brick & mortar outlets so I suggest buying an assortment of blades from that big on-line retailer.

I’ve just ordered this tool from Amazon but haven’t yet received it, so I can’t yet say how useful it is.