Will these table legs be able to support this table design or will they need additional supporting features?
Going by the additional detail provided in your Comment, I'm fairly sure that's enough support.
The "V" is secured to the table and bottom portion using two Biscuits at each joint.
This is your major design concern here, if I'm interpreting your intention correctly. It's important not to rigidly affix solid-wood tabletops to leg assemblies as it constrains seasonal movement — the wood of the tabletop must be free to expand and contract as its moisture content goes up and down through the year (a natural cycle, related to the humidity in the environment).
Instead of using biscuits it would be better to attach the tabletop to the cross supports using hardwood buttons (left), or more simply, expansion plates (right):
Note there are numerous other metal fastener options for attaching tabletops, I'm a fan of the above type as they are the simplest to install; they are also sold as "table stretcher plates" in case that helps in locating a source.