Is mathematics as pure as originally thought?
If I remember correctly, mathematics originally meant "the knowledge which is teachable", in contrast to the knowledge which can only be gained by experience. The emergence of formal logical systems like ZFC didn't worry mathematicians much in this respect. However, the computer proof of the four color theorem did really worry them, in exactly the way you suggest.
Both facts are easy to understand historically. (1) When ZFC emerged, physical computers where much weaker in their computational power than a skilled mathematician, so the computer only served as a logical tool for making the notion of "decidability" precise: "a statement is decidable, if and only if it can be proved or disproved in a finite amount of time by an idealized computer". (2) When the four color theorem was first published, it could be understood neither by a computer nor by a human alone. So it clearly was experimentation in exactly the way excluded by the original meaning of mathematics.
In a certain sense, the proof was simply not written down correctly, even so it was clear that it should be possible to write it down correctly, and find out whether it is correct or not. This has been done later, the original proof turned out to be incomplete, but it was possible to fill the holes. So the initial controversy was both justified and unjustified in a certain sense.
But today, physical computers have more computational power than a skilled mathematician, and the proof of the four color theorem has been formalized such that it can be understood and verified by a computer. Progress on the Riemann hypothesis has been in the form of computers checking that it holds for the first few millions zeros of the zeta function. The thing with the fully formal proof is debatable, but the computerized experimental verification of the Riemann hypothesis is clearly impure experimentation. For technical reasons, we have to accept that the computerized experimental verification of the Riemann hypothesis is mathematics, so mathematics is indeed not as pure as originally thought.
Is this not experimentation?
This is debatable. In a certain sense, you just repeated the very definition of decidability. This form of idealized computation is not yet experimentation. However, todays physical computers and their available software can actually decide many (challenging) statements, even if the procedure is not as stupid and straightforward as the idealized computation. But the intention of such an idealization is exactly to get a feeling for whether there might actually be a practical procedure as well, even if it should turn out to be much more complicated.