So did the British, where the
R in the Fairey Swordfish TSR and its predecessor, the Blackburn Shark TSR, and its successors the Fairey Albacore TSR and Barracuda TSR is for
Recon (TSR = torpedo/spotter/reconnaissance).
So, yes we'll need to change the spec and doctrine for the FAA's fighter, as there's no need for two Recon types on the same carrier, and especially one that degrades fleet air defence - ideally when the Nimrod replacement is being Specified. In 1940 each RN carrier should go to sea with folding Sea Hurricanes, Swordfish TSRs, and where there's space, Skua dive bombers (this also needs a
re-prioritization away from fighter pretences to dedicated DB).
Minor correction
S.24/37 that led to the Barracuda was for a "Torpedo Bomber Reconnaissance Aeroplane" combining the roles of torpedo bomber / recce with dive bomber / recce. Hence its designation became TBR not TSR. Later the Mark III version (with the radome under the aft fuselage) became the TR.III for torpedo bomber / recce. So effectively a successor for the Albacore as a TB and the Skua as a DB.
As for the Fulmar you need to sort out what it was originally intended for when ordered and what it was later
sometimes used for.
FAA fighters in the 1930s had two roles:-
1. Defence over the fleet, where the expectation was that enemy attackers would not have fighter escort; and
2. Strike escort to eliminate any enemy aircraft over the target & strafing (including the use of light bombs) enemy escort ships to reduce their AA protection. It is this part that meant a second seat was required as it would be operating out of sight of the RN fleet.
During the 1930s the FAA squadrons used a mix of single seat Hawker Nimrods (for part 1 above) and two seat Hawker Ospreys (for part 2). The Skua was expected to be the replacement for the second type, but development fell behind and it was delivered later than planned. The first was meant to be filled by the Roc turret fighter.
The spec to which the Fulmar was designed, O.8/38 called for an "Interim Two Seat Fighter" i.e. a stop-gap aircraft pending a decision about what should really be done. There was no mention of reconnaissance and recce was not part of its original function. The second seat was a requirement as it was expected that it would serve in the two roles as a fighter. The Fulmar was later used to some
limited extent as a recce aircraft, where fighter opposition was expected as it was seen as more survivable, or where its speed could be used to advantage. But it was limited. It was after the Japanese Operation C in April 1942 for example that Admiral Somerville's thoughts turned to using it in that role as his Albacores had proved vulnerable to the Zero.
1938 also saw an emergency purchase of Sea Gladiators due to a foreseen shortage of fighters in the FAA by the end of that year due to late delivery of the Skua / Roc and a diminishing number of available NImrod / Ospreys. It was effectively the successor to the single seat Nimrod as the fighter for protection over the fleet for use by C & G squadrons.
1939 saw the issue of new requirements but these did not result in any successful designs. Then right at the end of the year the RN reviewed its needs in the light of what happened since war broke out. From that in 1940 came a need for two fighter types:-
1. The two seater Firefly as successor in the two seat role; and
2. A 400mph single seater needed for base protection as the RAF couldn't do it as well as for use from a carrier. The Firebrand.
So there weren't two recon types on the carriers early war.
As for the air defence problem, you keep seeing it as giving the RN a better fighter. But it is a much deeper problem than that. The problem isn't doctrine. You are not recognising the detection problem. And you are not looking at the defensive / offensive balance of the carrier.
You have limited space on the carrier, any carrier. More fighters for self defence means less for strike escort and less strike aircraft, which is why you have the carrier in the first place. There is simply not enough room to carry enough fighters to maintain a full daytime CAP (even the USN & IJN carriers with their bigger aircraft complements only had 18 fighters to fill both the roles above). From the mid-1930s, with increasing aircraft speeds, the time between visually spotting an attacking force (and you can't guarantee that - see how the RAF Blenheims weren't spotted by the IJN in 1942 until after they dropped their bombs) had shrunk to such an extent that deck launched interceptors could not climb fast enough to intercept. There is no point in having a wonderful fighter if it can't actually intercept the attacker. Hence the adoption by the RN of the armoured carrier to protect its strike force.
1938 there is no radar. Radar changes everything. It pushes the detection point outwards. How far depends on many things, particularly height. Deck launched interceptors against low level attackers start to become viable. CAP becomes more viable as interceptions can take place further from the fleet giving DLI more time to climb to height, especially with higher performance fighters. But the problem of numbers still exists, including for strike escort.
1938 is an awkward year for the RN. Inskip has given them back control of the FAA in principle. But that doesn't become reality until May 1939. Going forward it can begin to train all its pilots to navigate to the standard of the RN pilots thay already have, so can
begin to think about dispensing with the second seat in future fighters. For a whole lot of reasons the Fulmar was what it spec called for, an "interim" fighter. And in 1938 radar at sea is still in the future, far less effective fighter direction.