With not needing subsidy and being open to private and licensed, you will be fine
Most daycares, private or licensed, won't know this far in advance if a spot will be available. Life changes and things come up too often to be able to guarantee a spot this early, IMO. I would suggest giving yourself around 6 months before going back to work to start looking. If a spot opens before you need it, you will (most likely) be required to pay for it to hold it and you wouldn't want to pay for months upon months I'd assume? This also gives you time to view as many daycares as you can/want to compare and get a feeling for the providers and routines.
There are LOTS of private daycares right now and I imagine this trend will stay true for awhile. Your options should be good!
Good luck and congrats on upcoming baby!! Oh and also, June is a good time to find a spot as many daycares have kids going off to school in the Fall and are looking to fill spot then.