if you use dovetails or finger joints, then the ends of the board will show, which wont be veneered. I think that that idea is a non starter.
If you were batch producing using a (purchased) veneered board, then I would look at an alternative joint- something like the lock mitre joint. You could then set up the router table which takes a bit of time, but then cut joint after joint knowing that the board is a constant thickness. With solid, I find it difficult to get a constant thickness to within fractions of a mm. The joint is very sensitive to thickness.
Personally, I would get a book with some designs in, and start by looking at how each is constructed. Andrew Crawford and Peter Lloyd are 2 authors that I would suggest- there are others too. Look at what will work for batch production, and the design options for each.
Another idea is to make your boxes and veneer them yourself. You could then add solid parts where required- where cuts show and where you will separate the lid from the base. It will be cheaper than solid, more stable, and you can use woods that you could not readily source/afford in solid.
You are looking to make 20-30 a week. That is approx 1 1/2 hours per box, which I think is unachievable. you are also trying to compete with mass produced items at that volume- veneered board and volume, it would be you vs somebody with a cnc setup, and there is only one winner, assuming that you could sell them. If there was a market, then somebody would come in and take it from you. In any case, you need to start by making a box, and step it up from there. Enjoy it first, because making a large volume regularly is entirely different to having an interest/hobby.