It varies. There's no exact yes or no answer, it can change based on so many variables. You can find them for cheap if you are lucky- I think I paid $200 for a 100% complete 1971-72, 76-77, 77-78, 78-79, 81-82 Topps and most of 1986-87 Fleer basketball sets plus some other stuff because the dealer just wanted to retire and be done with them. I've also seen people trying to get a couple hundred for a single card. Condition, location, it all plays a role.
By location I will give an example, last year I went to two card shows in Connecticut. One of them clearly favored the Boston teams and they were much more common...but also you're going to pay a little more for them. The selection vs. price is a tradeoff I was willing to make- I got my two oldest hockey cards there, both at the upper limit of my personally imposed single card price limit, ($20 per card, which I actually exceeded for them, paying $50 for the pair) and they were both Boston Bruins. The other teams were not available at any price.
As to rare...no. Nothing from the 60s or 70s flagships is truly rare. (speaking solely of basketball here). Paying for them all might not be easy- good luck paying for Kareem's rookie card from 1969-70 Topps, Pete Maravich's from 1970-71 or Julius Erving's from 1972-73- but you can find them. Those three I mentioned and the Magic/Bird combo from 1980-81 are the only real stoppers for that entire vintage Topps era.
Wait, let me correct that. The 70s inserts ARE rare. You will have a hard time locating some of those. Base cards though are always available, and usually in a variety of conditions.
If you go back into the 1957-58 and 1961-62 sets, those can be rare- there are some cards from both sets I've never seen in person despite being an avid collector of the sport since 1996. 1948-49 Bowman are genuinely rare, and even the base cards generally start at $20 each unless in really poor condition. All three of the pioneer sets have some seriously expensive stoppers, as well. The George Mikan card from 1948-49 Bowman sells for more than my parents paid for my house, for example. The 1957-58 Topps set is not rare- of the three pioneer era sets it's the most common, BUT...it also has a lot of HOF rookie cards. Bill Russell being the biggest, but most of the Celtics dynasty and many others. That pushes the set into high prices, but you could build roughly half of it on even a small budget. I've done that myself with 1957-58 Topps. Well, I'm at closer to a third of the set. The 1961-62 Fleer cards are much less common and usually found in lower condition...I don't know if Topps was made better or the Fleer cards in my price range are just on the lower end of the spectrum.
Again, this is just relevant to the NBA. I don't really have enough working experience with the NHL yet and I know nothing about baseball and football.
You could always do a "One from Every Set" project like I am doing with the entire sports that I collect. When that was a main focus (since supplanted by the "One of Every Person" project), I was basically pulling up a set on COMC and buying the cheapest card available. If you don't care who is on the card you can cast a pretty wide net. One thing though, to keep in mind, is that I stopped enjoing getting new sets that way in a very short order. It was nice to have it mostly completed, but when I found an example of cards from that set "in the wild", it took about 3/4ths of the joy out of it. That's why I'm specifically NOT buying them for hockey, now that I'm collecting that sport in full. I keep track of what I need, but I don't specifically seek it out. Yet...I may when I get down to needing only vintage stuff...and the NHL has a lot more vintage available than my other two sports, with sets going back to the 1920s.
Edit to add: nowadays with the internet you could probably find every flagship card for every sport within a matter of seconds. Rare is not really relevant when you have the entire world at your fingertips...my experince, other than on COMC, is all based on in person experience, at shows, shops and random stores and places that are not specifically card shops.
VERY slow trading due to health problems. Not transferrable so safe to trade with, just moving is painful and can't always access the cards.
Cardboard History My COMC
New Collection Website: Cardboard History Gallery (Still under construction)
Tips on how to make your scans look like the card does in hand (No more washed out, fuzzy scans!):