Is it better to use a microcontroller or pushbutton controller for simple on/off delay functions?
"Soft latch circuit", "soft latch power switch", or "pushbutton controller" are the keywords for searching for circuits like this. You can find tens of IC's that do this sort of thing by googling "soft latch power switch IC"; many have programmable delays.

I hate to recommend specific parts (cause shopping and buying recommendations are off-topic for this site), but I'll recommend STM660{0,1} and TPS342{0,1,2}. They're both parts I've used for similar purposes.


  What is the better option given that I intend to produce 1000s of these?


It's a tradeoff between cost, off-state power consumption, and manufacturing difficulty. You can get an ultra-cheap microcontroller (some manufacturers in Taiwan and China have $0.03 offerings), but it will take more power when the system is off and will add manufacturing and design complexity: now you have another chip that needs to be programmed when you manufacture the widget and another binary file to keep track of.
Generally, the specialized "pushbutton controller" chips will cost more (I realize that this is counter-intuitive, but I think it's because they are less general-purpose than an MCU and therefore have a smaller market), but will simplify design and manufacture and will be more efficient.