Your version of the TSP is actually NP-hard, exactly for the reasons you state. It is hard to check that it is the correct solution. The version of the TSP that is NP-complete is the decision version of the problem (quoting Wikipedia):

The decision version of the TSP (where given a length L, the task is to decide whether the graph has a tour of at most L) belongs to the class of NP-complete problems.

Your version of the TSP is actually NP-hard, exactly for the reasons you state. It is hard to check that it is the correct solution. The version of the TSP that is NP-complete is the decision version of the problem (quoting Wikipedia):

The decision version of the TSP (where given a length L, the task is to decide whether the graph has a tour of at most L) belongs to the class of NP-complete problems.

In other words, instead of asking "What is the shortest possible route through the TSP graph?", we're asking "Is there a route through the TSP graph that fits within my budget?".