The key papers to read about this are Vogel's who conjectures an even better 2-dimensional family which includes all simple Lie algebras, not just the exceptional ones. Deligne's family would correspond to a certain line in this plane. Personally, I find Vogel's papers difficult to read, but you have to read them if you really want to understand this stuff. Some good expositional sources are Cvitanovic's Bird Tracks and Introduction to Vassiliev knot invariants by Chmutov-Duzhin-Mostovoy. In particular, Cvitanovic developed some of the key ideas independently.
The basic idea is that you write down diagrammatically the universal metric Lie algebra object in a symmetric tensor category, using the category of Jacobi diagrams. This is very closely related to Vassiliev's finite type invariants by work of Bar-Natan. The resulting Lie algebra object won't be simple, but you can add the assumption that it is simple by assuming that certain Hom spaces are 1-dimensional. Roughly, conjecturally this added assumption gets you down to a 2-dimensional space of possible simple metric Lie algebra objects.
To get from the universal Lie algebra in the last paragraph to Deligne's exceptional family, you need to impose one more relation which is satisfied by the exceptional Lie algebras. Namely there is a certain invariant vector in the 4th power of the adjoint representation of any exceptional Lie algebra, and this gives an extra relation. This relation is written down explicitly on the first page of Dylan's paper mentioned in comments.
So where are we stuck? Well there's two problems:
- Do the relations that we can write down suffice to calculate the value of all closed diagrams?
- Are the relations consistent for all values of the parameters? That is, if you evaluate the same closed diagram in two different ways do you get the same answer?
The first of these questions should be easier than the second. But both are wide open.
What Dylan did in the paper mentioned in comments was look at some analogous situations and calculated directly that it appeared the answers to the first question in those circumstances was yes, but the answer to the second question in those situations was no, and in fact the relations imposed a polynomial relation on the parameter forcing you back to finitely many possibilities. But for Deligne's question, the calculation would require looking at very large examples which were beyond the computer's power.