The half-giant monster entry lists Powerful Build as a Special Quality. Alternate Form explicitly lets you retain those, so yes, a druid with Powerful Build does retain that bonus, which is sizable. Powerful Build explicitly stacks with things that change your size, so if the druid used Wild Shape to turn into a Large creature, he would count as Huge for the purposes listed in Powerful Build.
See here for a list of options for obtaining Powerful Build. Note that all of them have positive Level Adjustment, which the homebrew lesser half-giant you found does not. Many DMs would balk at that. In reality, Powerful Build on its own probably doesn’t justify LA +1, but Wizards evaluated it as such. Level Adjustment is bad in general, but truly awful for spellcasters. And the half-giant race has nothing else of use or value to a druid, though use of the Primordial Giant template from Secrets of Xen’drik, at least, would make the half-giant more caster-ly (though still nothing particularly relevant to druids).
There are guidelines for creating playable races in page 285 of the DMG, however, it doesn't go into detail on creating Large races, specifically (nor do they really go into detail on making medium characters, mechanically, save for the 'compare to other already-established races' approach). This is largely because playable races are only sized between Small and Medium (4 to 8 feet tall).
Take a look at the Goliath in the Temple of Elemental Evil player's companion, the Minotaur and Centaur race in their Unearthed Arcana version, and the final version of Minotaurs and Centaur in Guildmasters' Guide to Ravnica, all these races sound like they should be Large. In fact, if you take a look at page 223 of the Monster Manual, you'll find that Minotaurs are Large creatures, despite this, playable minotaurs are Medium. Of course, you could argue that the Krynn minotaurs who don't dwell mazes have evolved to be smaller. Then again, who's to say it wasn't a design constraint? ;)
So, why aren't there Large player-characters?
Weapon damage dice
In page 278 of the DMG, you can see rules on damage dice of creatures. A Large creature is going to wield oversized weapons, this means their damage dice are doubled, tripled for Huge, and quadrupled for Gigantic. A Large PC with a Glaive will be dealing 2d10+str, almost doubling the damage output just by being large, at 1st-level.
Area
A Large creature occupies a 10x10 ft area, making it more of a threat in battle especially when it comes to blocking and generally being a 'presence' in any encounter.
Aura Areas
As mentioned in this tweet by @ThinkingDM, auras emanating from Large PCs cover more squares than Medium PCs. Specifically, for a 5' aura, Large PCs cover 12 squares versus the Medium's 8 squares. In 10' auras, Large PCs cover 32 squares versus the 24 from Medium.
Squeezing
In order to accompany any typical adventuring party, a Large PC will need to squeeze in most places a typical adventuring party would visit. Whether it's a tavern, a forest gnome's tree house, or a mine tunnel made for dwarven miners, a Large PC will be continuously at a disadvantage while squeezing through most other terrain a Medium creature would have no problem fitting in.
These traits, inherent in the creature's size alone, make a Large creature fairly overpowered in most any game.
Best Answer
Not really, no.
Being size small has few, if any, direct benefits. Theoretically the small races (halfling, gnome) receive sufficient other benefits to balance out being size small, but it's primarily a sacred cow from 3rd edition (note that dwarves are not size small).
With minor situational exceptions:
Weapons with the Goblin Totem enchantment give an item bonus to damage against foes larger than you. Since most non-minion enemies are size medium or bigger, this becomes a very cheap way of getting a scaling item bonus to damage; it's much less reliable for non-small characters, since medium enemies are relatively common. Goblin Totem is useful for some builds (particularly relatively un-optimized ones), and despite being situational (the advantage over medium characters is only against medium foes) it's probably the best (if not only) benefit to being size small. Edit: There are other items/feats that provide a bonus against creatures larger than you, but not many.
You can move through the spaces of size large enemies (the rules say you can move through the space of any creature 2 or more sizes larger or smaller than you). You still provoke attacks of opportunity during this movement, but the potential positioning advantage over large enemies makes this one at least a little useful.
You can fit in smaller spaces without having to squeeze (assuming that's something that comes up in your game? maybe if you're fighting kobolds). By DM fiat, you may have an easier time taking cover, but there's no official rules to support this.
Small characters can use size medium creatures as mounts, but since mounted combat is usually a poor choice in 4e, and most of the methods of acquiring a mount that levels with you range from mediocre to atrocious from an optimization standpoint, the benefit of that capability is minimal.