To answer your question: "Do the pedals completely bastardize the sound the composers intended?" is not as much as playing on the wrong instrument does. :-)
The piano Mozart, Beethoven and their contemporaries would have played would have been a Fortepiano. The modern piano as we know it (named Pianoforte), came a bit later. he Fortepiano is a much quieter instrument. (Consider it this way: a pianoforte can be heard over a full Romantic period orchestra, a fortepiano really can not.) [There also existed an even quieter instrument, the clavichord. It was mostly used for practice and very small, intimate recitals.]
Both the pianoforte and fortepiano use the same basic concept of striking a string, instead of the picked string of the harpsichord, thus allowing for varied dynamic range. [The harpsichord basically has only one volume level for each key. Changes in the overall volume in a harpsichord piece were often done by layer the number of notes.]
Having said all that, the wonderful thing about music is that it is open to interpretation and reinvention. When I listen to music, I listen more to hear what the artist brings to the music than I do to how perfect a reproduction of exactly was was written on exactly the right instruments.
In the end, it's up to you to decide what sounds best in your interpretation.