Absolutely. There was a great deal of interconnection between musical eras. There are so many directions I could go with this so I'll go with some of the names that you mentioned.

Did the people in Tchaikovsky's time enjoy Bach or Beethoven

Tchaikovsky himself had a great deal of respect for Beethoven. In a letter to a student, he wrote, "I cannot think of any musical works (with the exception of some by Beethoven) about which one could say that they are completely perfect." This is from letter 517 and you can see a translation here..

Now, that's Tchaikovsky and you did ask about "people", the challenge being that we don’t often know what many common folk thought.

But we know what was publicized and what was performed. So here it is worth mentioning that Beethovenfest has been held in Bonn, the city of his birth, since the 1840s. You can also find in the historic archives of various symphonies around the world evidence of Beethoven performances in the late 1800s. For example, in Pittsburgh in 1898, the Pittsburgh Symphony Orchestra programmed Beethoven's Leonore Overture No. 3.

Likewise, did people in Beethoven's time enjoy Handel and Vivaldi?

Again, most certainly. In fact, it has been said that Handel was Beethoven's favorite composer.

Bach, who you mentioned above, is an interesting example as well. He wrote so much for the keyboard, and after his death several of his students and a few of his sons began transcribing his piano works and passing them along to their own students, who pass them on, and on, and so forth. Thus these pieces endured because it wasn't just people going to listen to them - they were being played on pianos in private homes throughout the region.

This includes pianos touched by Beethoven. In an issue of a music magazine from the early 1770s, you can find this quote which is speaking of Beethoven:

...a boy of eleven years and of most promising talent. He plays the clavier very skillfully and with power, reads at sight very well, and — to put it in a nutshell — he plays chiefly The Well-Tempered Clavier of Sebastian Bach...

So in sum, yes. Composers were well aware of their predecessors, and the idea of symphonies playing more than just new compositions is not a novel one and stretches back through the various eras of music.

Again, there are so many ways to look at this question so if there's something more you want me to dig into please let me know. The answer might seem simple and surface but the question seems to be as well. I'm but an amateur historical musicologist so if there's an area ripe for expansion please say so.