Fixed parameter tractability is not necessarily about polynomial time algorithms with huge constants; it's about getting efficient algorithms for problems that have different parameters. Optimising over all the parameters might require exponential time, but if you fix one of the parameters, some problems become tractable (with the constant depending on the value of the fixed parameter).
This can lead to large constants, but doesn't have to. It depends on the value of the fixed parameter.
In this case the polynomial time algorithm is to test the graph against a finite list of forbidden minors. This has a reasonable exponent, but a constant dependent upon how long that finite list is.
That finite list can be extremely long indeed. And in some cases, there is no way of finding or verifying the full list.
This can lead to large constants, but doesn't have to. It depends on the value of the fixed parameter.