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I'm just guessing here but I would think that cooking would kill off anything picked up from the thawing process.
It's nasty, but not necessarily unsafe.
And now you're washing your dishes in a sink coated with whatever came out of the meat
I mean that's just as true as if you let the meat defrost in a plate and then put that plate in the sink.
Not necessarily toxins (as in ones produced by bacterial/fungal processes) or toxic/bad-tasting chemicals left over from cleaning dishes.
Assuming dishes have been cleaned and not just rinsed/wiped
If they're cleaned in a sink that just had raw meat floating around, they're not clean regardless.
Do you rub your dishes on the walls of the sink when you wash them?
No, but I usually wash them in water that's been touching the sides of the sink