Unfortunately, without trying it myself I couldn't say if the control would be good or not. There is too much variation in the implementation of these products, if it has IFTTT then it is much more likely to work with C4, but this requires additional drivers which are licensed.
But, if Control4 can tell the blinds to open / close then yes it will be able to do all the scheduling and sunset / sunrise control. You wouldn't need to touch the Tuya app.