The problem is even a little broader than this. Lulu is not the only experienced guardian - Auron served as Braska's guardian and got even farther, reaching Yunalesca. Auron's starting HP is 1030, which is definitely more, but he is a fighter class, so "more" is to be expected.
I would like to offer some potential explanations.
- No Sphere Grid
Perhaps both previous efforts were "No Sphere Grid" runs, where the player intentionally does not increase any of the characters' attributes. It is certainly possible to make it to the Calm Lands while keeping Lulu at 380 HP.
- Lulu's Broken Heart
Lulu is a mage and magic is fundamentally a mental ability. A spellcaster has to "have one's head in the game," so to speak. We know that Lulu is grieving over Chappu, and Tidus resembles Chappu, so she may be (at least initially) operating under significant psychological duress. (Auron is also laboring under the effects of trauma, though his is physical.)
- Tidus the Rock Star
As we learn on the Mi'ihen high road, ordinary citizens of Spira are able to travel about even in the presence of fiends. But when the party is informed by the Chocobo Knight of "a large fiend" in the area, Tidus' reaction is to find it and engage it. This is a surprise to the rest of the party (except for Auron, who recalls that Jecht had a similar attitude.)
It would seem that "standard operating procedure" for summoners is to "blend in" with passing traffic, and try to avoid fiends. This strategy would result in characters earning little or no AP, which is consistent with Lulu's low starting HP.
(Jecht's similar "rock star" attitude may be part of why Braska was able to make it as far as Yunalesca.)
- Jecht the Stalker
Most summoners have to seek out Sin once they have undergone the Final Aeon ritual. But Jecht seems to be following Tidus around. This results in several "boss" battles facing Sinspawn, and even Sin directly, which are a major source of AP for the party that would not have been present in Lulu's previous pilgrimages.
- Yuna's ADHD
We run into other summoners who seem very focused on getting to their destination. But Yuna seems to have a problem focusing - getting kidnapped (twice), starting a civil war, and even getting married. So instead of following a direct path from temple to temple, the party ends up chasing her all over Spira.