Titanium frames have a devoted fan base. Personally, I tried to notice the difference back when I did road racing but didn't. Maybe I didn't want to: Ti ain't cheap.
Low quality steel is very cheap but also really heavy. You can get a whole crappy bike for a couple hundred bucks. Higher end steel isn't as cheap but is still relatively heavy (compared to aluminum or carbon). You can get this kind of bike in the $1000-2000 price range (e.g. Surly). Aluminum bikes tend to be inexpensive, but also not the lightest. These can also be priced at $1000-2000 (Specialized, Trek, Giant, ...). Carbon comes in a range of prices with various tradeoffs. You can get very budget carbon frames at like $1500 (Winspace) or whole bikes at like $3000 (Giant) (maybe $2000 in non-major brands, very discounted during current market conditions).
For titanium, I'm seeing Black Friday deals starting at like, $3200-3500 (Lynskey / Litespeed). But they're often sold at higher prices than carbon bikes. (For my money, I prefer carbon frames -- you get more flexibility in tube shapes and the end result can be lighter and stronger than titanium.)
For about the last year and a half you can usually get a carbon road bike from major brands with basic components starting from 2000 euro (right now I see Bianchi, BMC, Cannondale, Cube, Giant, Specialized listed around that price).
Yeah, I think in particular prices have come down dramatically in the past 3-6 months.
The specific 2000EUR figure might be European pricing. The lowest I'm seeing for any carbon road bike from e.g. Specialized with US pricing is $2400 for a two generation old Tarmac SL6 (this generation was current 2019-2021) or $2800 for a current Roubaix. The least expensive carbon road bike offering from Giant is $3300 US (TCR or Defy). (Though I'd expect to be able to find better deals at retailers trying to move old inventory than direct from the manufacturers.)
Yeah EU. One example I can give is this, Specialized Aethos 2024 for 2200 but only one size available, Giant TCR Advanced 2024 several sizes 2400, plenty others. This is a reputable retailer and there are some other with same/lower prices.
Prices shot up during the pandemic and never really went back down. You're lucky to find a good carbon bike for under $4K these days where you used to be able to find them for $2K.
Quality used bikes are remarkably affordable. I'm a sucker for 80s road bikes; $200 gets you something decent, bring it to a shop for a tune up and probably new brakes.
10-speeds are most common, so you have to be ok with that; although some vintage bikes did have 3 chain rings. Older road bikes tend to have narrow tires, because everybody thought they were faster; 1/4" tires are workable on unpaved mixed use trails, but 1/8" will be pushed around easily by tree roots and ruts.