You can exile Stroke of Luck even if you draw the symbol. The test is still considered a failure, but stroke of luck is exiled. The ability to exile Stroke of Luck is not dependent on the auto-fail not being revealed - only the test being automatically successful.
Automatic Success/Failure: Some card effects make an investigator automatically succeed or automatically fail a skill test. If this occurs, depending on the timing of such an effect, certain steps of the skill test may be skipped in their entirety.
- If it is known that an investigator automatically succeeds or fails at a skill test before Step 3 (“Reveal Chaos Token”) occurs, that step is skipped, along with Step 4. No chaos token(s) are revealed from the chaos bag, and the investigator immediately moves to Step 5. All other steps of the skill test resolve as normal.
- If a chaos token effect causes an investigator to automatically succeed or fail at a skill test, continue with Steps 3 and 4, as normal. - FAQ, v.1.7, March 2020
The exile mechanic is an interesting one. Cards with an exile mechanic are extremely expensive near the beginning of a campaign but seem reasonable near the end. If purchasing before the last scenario, any card you purchase has an "exile" effect.
Stroke of Luck came out with the 2nd to last Mythos Pack of this cycle. So what should be the XP value of this skill card? Most skill cards have an XP value of 1 with few exceptions. This card however, is much more powerful than any skill card. No skill card can do better than Stroke of Luck. It's the most powerful skill card in the game. You could have a base skill of 0 and still have a 95% chance of beating a skill check of 10 or more. And as such, it should have a higher cost than other skill cards. But how much more? Double? It's probably worth a bit more than double considering it's Wild and its also saving you from using multiple cards on one skill check. Three times? Depends on the scenario. Will there be those situations where you need a boost of +4 or more? In the last few scenarios it's much more likely.
The conclusion is that Stroke of Luck seems like its very powerful for the final scenario. It only costs 1 more XP than any other skill card and the exile drawback is negated. For the second to last scenario is its breakeven point. I wouldn't recommend purchasing Stroke Of Luck prior to the second to last scenario.