For Djikstra's algorithm, why are we surely done if we update all edges $|V|-1$ times? Apparently, if we use Djikstra's algorithm to find the shortest path between the root node and all other nodes in a weighted graph with no negative cycles, we are done after updating the distance of each node $|V| - 1$ times.

This puzzles me because I think that a single round of breadth first search is enough. Why must we do $|V| - 1$ of these searches?
Either you mixed up two algorithms or you misinterpreted an upper bound. One level of a BFS with priority queue is enough, but during this one round a neighbor of the root may be updated $|V|-1$ times, therefore the update operation needs to be efficient.