I've only brought job motivation down to zero once but yes product drop became zero.
that's not correct. even with 0% motivation you can still get products.
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maybe you did 15 sec jobs at 0 motivation, where the chance to find a product is just 2-3%, and that's why you didn't find any.
regarding luck drops - try it and tell us
I use motivation buffs or move to a different job if the motivation is low and I can't tell you first hand what happens at 0. I wouldn't waste my time and energy on something that gives no money and experience just because there might be a chance to find some random item. if you're looking for recipes on Lead a trading office or similar jobs (that's what most people would do to get their motivation to 0, I guess?), maybe you should stock up on some motivation buffs first.