It that Professor Oak actually studies

The stats for a Pokémon are based on:

• The base stats
• The IV for a Pokémon (depends on the one you caught)
• The EVs gained while training a Pokémon (depends which types of Pokémon you battle)
• The nature of a Pokémon

The relevant formulas are:

and

for the first and second generation, and:

and

for the third and fourth generation.

As you can see, nowhere is there any mention of having a gym badge in these formulas.

The theory as to why the game tells you there is a stat boost that, in my opinion, holds the most ground, is that the EVs you get from battling the gym leader's Pokémon match the stat that they tell you gets raised.

Source for theory
Source for stats

False Swipe leaves an opponent with at least one health. In Leaf Green, the following Pokemon learn it by levelling up. Entries in bold are Pokemon which can be caught in Leaf Green; others can be acquired by trading:

• Cubone (level 33) (Sevault Canyon, Pokémon Tower floors 3-7)
• Farfetch'd (level 46) (Trade a Spearow to a girl in Vermillion City)
• Grovyle (level 53)
• Marowak (level 39) (Sevault Canyon, Victory Road floors 1-3)
• Nincada (level 25)
• Sceptile (level 59)
• Scizor (level 16)
• Scyther (level 16)
• Zangoose (level 55)

The following Pokemon can hatch with it as an egg, if their father had the move:

• Heracross
• Mawile
• Paras
• Pinsir
• Seedot
• Shroomish
• Spearow