Can you be sent to Reading Railroad without passing Go?
Yes, it can happen...
There are 2 cases that explain the wording of this card. These cases reveal that it is possible to be sent to Reading Railroad by a Chance card from a square other than a Chance square.
Case 1 Original USA game
According to World of Monopoly, these (first?) four versions of Monopoly contained this Community Chest card : «Pay $10 fine or take a Chance»

Darrow White Box, Darrow Black Box, Parker Bros 1935 Plain, Parker
Bros 1935-36 Plain


This means that you could be sent to Reading Railroad by a Chance card from the Community Chest square between Mediterranean Avenue and Baltic Avenue, and NOT pass Go.
According to EnAcademic.com, this Community Chest card has been omitted from the US editions since 1936.
Case 2 Blame Canada
Again, according to World Of Monopoly,

Edition:  “Mini box”, with separate board - Atlantic City Publisher:
Parker Bros. Inc/The Copp Clark Co.Ltd. - ±1946 ... Reg. U.S. Patent
Office Canadian Registration M.S. 19-5674 U.S.  Patents 1.509.312
---2.026.082  Patented Canada.1936 ---- 362.124 ...
«Curiously, the rules of this game (in English and French) contain a
provision that a player rolling a 3 may take a Chance or Community
Chest card instead of making his move.»

I remember playing with this rule in the 1950s, but I always remembered it as a house rule. It appears that it was official, in Canada.
Therefore, if you rolled a 3 while on Mediterranean Avenue, Community Chest(1), Baltic Avenue or Income Tax, you could elect to not play it, take a Chance card, and be sent to Reading Railroad by that card, without passing Go.
Conclusion
In either case, it would be possible to be sent to Reading Railroad, by a Chance card, from a square other than Chance, and not pass Go. And that is why the phrase «... if you pass GO, collect $200» was required on the Chance card in those days. The phrase just was not dropped when no longer required.