Were tongue twisters meant for fun? Or were they intended for improving speech?

Solution 1:

It appears that tongue twisters were originally invented for fun, and were later used for improving speech articulation: Britannica.com:

• Tongue twister, word or group of words made difficult to articulate by a close sequence of similar consonantal sounds. Tongue twisters are often passed on for generations, becoming a rich part of folklore. Two widely known English-language twisters are “She sells sea shells beside the seashore” and one beginning “Peter Piper picked a peck of pickled peppers.” Some are more difficult to enunciate—“The sixth sheik’s sixth sheep’s sick.”

• Tongue twisters have been recommended for curing hiccups and for curing lisps and other speech defects. They are also used for testing the fit of dentures and for screening applicants for broadcasting positions.

The is no mention in Etynomline of their origin as a medical 'device': Toungue-twister (n.):

• 1875, in reference to an awkward sentence, 1892 of a deliberately difficult-to-say phrase, from tongue (n.) + agent noun from twist (v.). The first one called by the name is "Miss Smith's fish-sauce shop."

Origin of two famous Tongue Twisters

• The tongue twister “she sells seashells by the seashore” was inspired by a woman named Mary Anning, who spent her time gathering seashells from beaches to sell to tourists.

• "Peter Piper picked a peck of pickled peppers” also had a real-world origin. There was an actual person whom Peter Piper was based on: a French pirate and horticulturalist named Pierre Poivre