I am having trouble finding the quality of old telescopes. My question is how does a decent modern telescope that cost about a couple hundred USD compare to the top-notch telescopes in the nineteenth century. Can one make the same observations now with such a modern telescope as one could do in the nineteenth century with a top-notch one?
Modern telescopes, even those that can be purchased from stores for, say, under $10000 (which is already a lot), were produced with computer-controlled equipment to polish the mirror and make the lenses and mirrors, and the lenses usually have nano-coating to reduce reflections and increase light transmission.
Overall you have a century of progress & experience building them since then. Mounts are better as well, mechanically and electronically (you have auto-guiding whereas it was manual a century ago, with an astronomer watching a star through a reticle). In other words, modern telescopes, even amateur ones, are vastly improved.
You can compare photos (from various books such as https://www.amazon.com/Day-We-Found-Universe/dp/0307276600) to modern astrophotography done by amateurs (eg https://www.astrobin.com/explore/top-picks/). The modern ones (also benefiting from sensors more sensitive than film) are much better.