So, this is slower than the seqscan, which means the planner made the right choice.
You seem to be imagining that there's some way the index could be used with the NOT clause, but there isn't. Indexable clauses are of the form indexed_column indexable_operator constant and there's no provision for a NOT in that. If we had a "not contained in" array operator, the NOT could be folded to be of this syntactic form, but it's highly unlikely that any index operator class would consider such an operator to be a supported indexable operator. It doesn't lend itself to searching an index.
So the planner is doing the best it can, which in this case is a full-table scan.
A conceivable solution, if the tags array is a lot smaller than the table as a whole and the table is fairly static, is that you could make a btree index on the tags array and let the planner fall back to an index-only scan that is just using the index as a cheaper source of the array data. (This doesn't work for your existing GIST index because GIST can't reconstruct the original arrays on-demand.) I suspect though that this wouldn't win much, even if you disregard the maintenance costs for the extra index. The really fundamental problem here is that a large fraction of the table satisfies the NOT-in condition, and no index is going to beat a seqscan by all that much when that's true. Indexes are good at retrieving small portions of tables.
