I've seen discussions on this site about adding wood ash directly to the vegetable plot. But I was wondering if it would be better to add it to my compost bin instead?
You can add wood ashes to compost piles, but only in small quantities, because ashes are alkaline and will increase the ph of your heap. Either add some and mix them in well, or add a thin layer every few inches as you build the heap - the ashes shouldn't be so thick you can actually see them as a distinct layer once added. There's some guidance here regarding use of wood ashes generally, including in compost - you can certainly use it direct in the garden, but again, only in small quantities, and not around acid loving plants like rhododendron https://www.rhs.org.uk/advice/profile?PID=621
Speaking from direct experience, having lived for 20 years in the Canadian woods and heated entirely with a woodstove, it really depends on how much ash there is and whether it has been aged/weathered.
On a number of occasions I have destroyed entire rows of crops in the garden by applying ash that is too fresh too thickly. I thought pieces of rhubarb might send out roots with a little ash in the soil but no go. Too much chemical activity.
My current policy is to put most of the ash directly on my grass plots, allowing the rain to flush the nutrients into the ground for the benefit of the grass and weeds, collect all grass clippings and put the clippings on the compost. Never a problem with this very cautious method which produces very healthy grass and lots of compost. A small portion of the ash goes directly under the fruit trees which have strong root systems and can benefit from the potash. In both cases the ash is spread somewhat thinly, avoiding big dollops (technical term) of ash.