The question of pedal in Mozart and Beethoven does not have an easy answer. It's a complex problem with many factors weighing in.
First, the pianos they had were very different from ours. It did not have an iron frame; the strings were thinner; the hammers were smaller and covered with leather instead of felt (in Mozart's time). These instruments had very little power and sustain compared to today. They were fitted with pedal. Mozart used pedal, but we don't know how, when, how much etc.
With Beethoven came a new era of pianoplaying. He himself said he thought Mozart's style was 'dry' (he heard him play). Beethoven was a genuine legato player, and had pianos fitted with even more pedals than we have. Sometimes his music contains the instruction una corda (=one string).This is because in his time you had pedals to shift from three (or four) strings to two or one string (on the same note). Alas, that instruction meant pressing down one particular pedal he had that shifts to just one string, creating a very hushed sound. Today we simply press the left pedal, in reality shifting from three to two strings.
With the right pedal, the sustain pedal, we also enter a new era with Beethoven. There's the famous instruction he wrote for the Moonlight first movement:
"Si deve suonare tutto questo pezzo delicatissimamente e senza sordino"
=You must play this whole piece delicately and without dampers.
That means, press the pedal down and never lift it! Arguably, nobody ever asked for so much pedal.
Still, we are left with little information. It's not until the next generation, Chopin, Schumann, Liszt, that the pedal becomes a truly integral part of the structure of the music. They also wrote in great detail in the score about where to use it.
Then came the 20th century. Instruments of enormous power and sustain. A new pedal-technique emerged to enhance the clarity, basically
1.hit the note
2.press the pedal
3.lift the hand
4.hit the next note and simultaneously lift the pedal
Also emerged the 'style' of
no pedal in Mozart and Bach
a little pedal in Beethoven
ignore Chopin's pedalmarkings
All this was news.
Back to Beethoven and Pathetique. It needs a lot of pedal. But sometimes teachers will instruct you to practice without the pedal in order to master the balance and phrasing with your fingers alone. Also, the pedal needs to be applied with great sensitivity, for when we use the pedal we must never loose control of chord voicings etc.
If Beethoven was able to hear a modern performance with only a little use of the pedal, chances are he would say "you make my music sound like Mozart's".