I have built a collection of 3CS in XF over about 3 years (five elapsed now). You are correct, 1851-1862 are easy to find, but take your time and pick nice ones (there are a lot of really ugly 3CS out there).
1863->1872 are basically impossible.
For both business strikes and proofs, mintages are very small and very, very few coins have survived. In 1874 the mint melted vast quantities of obsolete coins. Even into the 1910-1920 period large numbers of 3cs were turned into the treasury and melted.
You also need to read between the lines of the census reports at the TPGs.
For example, 1863
Mintage: 21,000
Certified NGC: 78, lowest grade AU55
Certified PCGS: 105, lowest grade XF40, 10 in AU
PCGS survival estimate
All grades: 500
60 or better: 200
65 or better: 100
XF Auction Sales: None
Or 1871
Mintage: 3,400
Certified NGC: 141, lowest grade VF (q=3), 1 XF40, 3 XF45
Certified PCGS: 124, lowest grade VsF (q=3), 1 XF40, 4 XF45s
PCGS survival estimate
All grades: 800
60 or better: 250
65 or better: 125
XF Auction Sales:
XF40, NGC, 2011, $1,093 (Heritage Auction)
XF45, NGC, 2016 & 2013, $940 (Heritage Auctions)
Price guide XF40 - 690, XF45 - 700 --- bunk
The point being, when they tell you that the 1863 is worth $525 in XF, that's pure fiction. It doesn't exist. It's never been sold at auction. PERIOD.
In the last five years, a dealer I know has bought over the counter, certified and sold four 1871s in XF (one was details). I was a good customer, with cash, and I know I didn't pay $700...
PS: Subtract those coins from the census and you'll see what existed when I started out...
So figure you are settling in for a long slog to finish the set.
Can it be done? Sure. Just time (lots) and money (lots).
To get to where I am, I've filled a couple of the holes with impaired proofs, which I may never upgrade... and I'm happy to have it.
Besides, there
is something fun in owning a coin described as
1872 3CS PR50 NGC. NGC Census: (1/201). PCGS Population: (0/286). Mintage 950. (Registry values: N1) (NGC ID# 27CH, PCGS# 3723)
i.e. 1 at this grade, 201 higher (they don't quote the 0 lower

)
References:
https://www.NGCcoin.com/coin-explor...coinid-13682http://www.PCGScoinfacts.com/Coin/Detail/3682https://www.PCGS.com/pop/detail.aspx?c=77&p=MS&t=5
-----Burton
50+ year / Life / Emeritus
ANA member (joined 12/1/1973)
Life member: Numismatics International, CONECA
Member: TNA, FtWCC, NETCC, EveryCountry (online) coin club
Owned by three cats and a wife of 40+ years (joined 1983)
Author: 3rd Edition of the Sample Slabs book,
https://www.sampleslabs.info/