I wouldn't go so far as "esteemed", lol.... Wait 'til you get to know more geologists...
A) It's not just carbon-dating, it's radiometric dating, using all sorts of things...
Carbon-14, Pb-Pb, K-Ar, etc. etc. etc. And, it's cross-checked by multiple iterations, against other types, not just depending on a single hit... (FWIW, Carbon-14 is only useful for "recent things", less than ~50,000 years... you have to use the lead or potassium-argon to get into the really old dates.)
B) Whenever I'm dealing with someone that's cutting it off at around six thousand years old, I realize that I could talk/type 'til I'm blue in the face about radiometric dating, so I use a different tact, and bring up the glacial varves.
If you look at areas near glaciation, you will find in the sediments below the glaciers where glacial deposits are being made. When the deposits are being made into a fairly still body of water, you will see an alternation between the summer and winter layers that distinctly shows up in the sediment as alternating layers of colored sediment, in light and dark layers (actually, it's coarse and fine that is alternating, but is usually more noticeable by color). If you take a light and a dark layer together as a couplet, you've defined a year's worth of sediment. Start with this year, take the top two, the next couplet is from last year, so on and so forth. So, you can start from today and go back through time counting couplets, counting years. Sweden is where the idea came about, and have places that go back 15, near 20,000 years, but there are lakes in Japan that go back from today to near 50,000 years ago. That alone has thrown a "young Earth" out the window. But knowing how this pattern works, and identifying fossil varves in the rocks, we've seen 100,000 year stretches over and over at many different points in the Earth's history. So, if we had a couple of hundred thousand years here, and a couple of thousand years there, etc. etc., we quickly end up realizing that, even if we don't look at anything but varved rocks, that we've got to have several million years to make up those layers alone.
Now, think about all the layers of beach sand, and sea shells, and mud flats, and so on and so forth.... LONG before anyone had any idea of using radioactive decay to come up with actual numbers to the age of the Earth, geologists had used the principles of "cross-cutting relationships" and "superposition" and "faunal succession" to figure out the order of deposition of most of the rocks you see. They realized, to have seafloor get crunched into the middle of a mountain core, then erode most of it away, rift it all apart, then crunch it back up in the middle of a bunch of newer seafloor, had to take a phenomenal amount of time... hundreds of millions of years.... and that's just for Phanerozoic Eon, with fossils involved... once you get into the PreCambrian, you've got many more times the amount of rock and layers to deal with, you just didn't have the fossils there.....
