Why would a Bosch dishwasher start leaking out of nowhere?

Yeah, that seam can just let go on its own around that age. Bosch tubs are plastic and they sit on a lower base with a seal between the two, and after enough years of heat cycles that bond dries out and lets go. The crusty stuff is a giveaway too, that's detergent residue left behind when water seeps out slowly and evaporates before it ever reaches the floor in any real quantity. Nothing you did wrong, it's just age.

Had nearly the same thing on mine a few years back. Before you assume the worst, pull the machine out and check it with a level. These floor seal issues sometimes show up when a foot wanders or the floor itself settles, and the tub racks out of square. Mine had tilted just enought after I replaced the kitchen floor and never shimmed it right.

If it realy is the seam itself, an appliance tech can usually reseal it in one visit. Back in the day we'd clean the joint good and lay a bead of appliance-safe silicone, let it cure a full day, and that held for years. Cheaper than writing off a machine that still washes fine.

Seconding the level check, and I'd add that where the water shows up isn't always where it's coming from. On mine the drip turned out to be a weeping connection partway up, and it ran down behind the panel before pooling at the front like a base leak. I only found it by running a cycle with the side panel off and watching with a flashlight. Took about twenty minutes with a screwdriver and saved me from chasing the wrong thing. Might be worth doing before you assume the seam itself is gone.

One thing I'd check while you've got the toe kick off is wheather any water has actually made it into the base pan itself, becuase that's where the float switch lives on these machines and it doesn't take much standing water in there to trip it and shut the whole thing down mid-cycle. Mine never leaked enough to reach the floor but eventually the float did its job and the machine just quit with an error, which was honestly the most useful diagnostic it ever gave me. If you find moisture in there, dry it out thoroughly with a towel before you run your next test cycle, otherwise you'll be chasing a ghost from the previous wash.

Update: the level check paid off. Pulled the machine out and the front left foot had backed off a bit, so I re-leveled it and the pooling is way less after two cycles. Still a little seepage at that seam though. Does the seal between the tub and base ever reseal itself once things are square, or is that a replace-it-once-it-leaks situation?