Why couldn’t Jar Jar, Qui-Gon, and Obi-Wan go around the planet. The core would only be the fastest if they were on the opposite side of the planet. This wouldn’t make sense, as they can’t have travelled far from where the droid army (source: still appears to be the same time of day, although I could be wrong). Is this just Boss Nass hoping to get lucky and indirectly kill Jar Jar (and by connection Qui-Gon and Obi-Wan) or was it legitimately the shortest route, and the Trade Federation just landed on the opposite side of the planet?
The 'Inside the Worlds of Star Wars: Episode One' factbook tells how eons ago the last remnants of "plasmatic magma" moving around inside the planet formed deep and long waterways in the bedrock, before finally dissipating into the rock itself. These are the underwaterways that the Gungans refer to as the "planet core". They don't strictly speaking travel through the entire core (e.g. from one side to the other) but they do burrow into the core material.
The Jedi basically went down a few miles, then travelled along (sideways), then came back up a few miles when they got to Theed, which as you've noted isn't that far away from where they landed.
Underneath the surface of Naboo, a tremendous maze of passages and caves, created by movements of unstable plasmic energy in the interior, provides the Gungans with transport routes through the planet. Navigating these underwaterways requires immense skill since they are home to ferocious creatures. A single wrong turn can spell certain death. In spite of the risks, fleets of Gungan trading subs constantly voyage through these routes; overland travel on fambaas or kaadu is slower and more arduous for this amphibious species.
As to why they thought that this was a good idea, this is addressed in the film's official junior novelisation. The Jedi previously had no transport at all and while this bongo might be crappy and dangerous, it's still better than Shanks' pony.
The bongo the Gungans had promised turned out to be a tiny, bat-winged submarine with three bubble canopies and a strange drive that looked like long, trailing tentacles. Obi-Wan eyed it dubiously, but it was better than walking. Or swimming. He slid into the pilot’s chair.