Right, except that it's not. If the DM can call for an ability check, he can use whatever ability he feels like using without it being either a house rule or a home brew.
There is no rule that they can show me that says that, though. There is only the general rule for calling for ability checks, which gives the DM the right to use any ability he chooses. There is no rule that says that it's okay to use con, but not strength for climbing and swimming.
Personally I agree with you, because I think that the rules leave it to the DM to decide what complications are close enough to the examples in the book to call for a check. So if a DM thinks that a given complication (such as height of climb or length of swim) is sufficiently close to the Strength examples, they can call for a Strength check, and if the DM thinks that a given complication is sufficiently close to the Constitution examples, they can call for a Constitution check, and that if the DM thinks a given complication is sufficiently close to the Strength and the Constitution examples, the DM can call for either.
But this is the same fundamental difference that has divided the thread from the beginning. @Charlaquin and @iserith instead read the examples more narrowly, and require the DM to only call for a Strength check if the complication is sufficiently similar (according to an unstated standard of correctness) to the Strength examples, and similarly for Constitution. From their standpoint, the rules do not permit the DM to call for a Strength check unless the DM correctly determines that a given complication is sufficiently similar to the examples.