Reduction to longest path from longest chain of tiles
But I wonder, I have shown that in general you have to solve the longest path problem to find the longest chain of such tiles


Not exactly, you have given a way of finding the longest chain of tiles (transform the tiles into a graph and solve longest path). There are many other possible approaches (you can try to transform it into a boolean formula and solve a SAT instance).... how do you know that none of them is easier? You could also use undecidable problems to solve your tile question (build a turing machine that halts iff the tiles have a solution of some size), it doesn't mean that your problem is undecidable.

Reductions work the other way around. I like to see an NP-hard problem as "a tool that may be used to solve any other NP problem". So  you need to show that your tile problem is one possible tool for solving longest path. That is, you are given a graph, and you need to describe a set of tiles with the good property (the tiles have a length-k solution iff the graph has a length-k path). This way, you know that your tile problem is a new possible tool for solving longest path, and, in turn, every other NP problem.

PS: there are also errors in the reduction you describe, beside the method problem discussed here: if the same integer appears in three different tiles (say, 4-2,4-7,4-9), then theses three nodes would form a valid path in the graph, but not a valid sequence of tiles. You'll find it easier to see a tile as an edge rather than a vertex