Doran's shield sucks and nobody uses it because it has weak passives instead of stats.

First of all let's compare the Doran's items to make it plainly obvious how different Doran's shield is.

Doran's Ring: HP + AP + Mana regen + MP per kill. All of these stats, including the regen, scale throughout the game. Doran's Blade: HP + AD + Lifesteal. Again, all of these stats scale throughout the game. Doran's Shield. HP + 1.2 health per second + 8 damage reduction from AAs and single target spells.

So without even going into the extremely flawed passives, you can see that Doran's shield offers practically no flat stats. Doran's has no secondary stat (AP/AD from the other Doran's items), which would be AR or MR or even more HP than the other Doran's items, and it's missing a regenerative stat such as 50% HP/5.

Now onto the passives. 6hp/5 is very weak. Most fighters and tanks have 8+hp/5 at level 1, and are around 15hp/5 or higher by level 9. When you look at the gold efficiency of other Doran's items, you could rationalize that 75-100% HP/5 would be an appropriate amount of %hp/5. This means that at level 1, most fighters and tanks would have close to the same HP/5 value that Doran's Shield currently offers, and by level 8 they would have more than double the HP/5 from a % value than the flat 6hp/5. So this shows how horribly the shield scales and how turning the flat hp/5 into a % hp/5 would solve one of it's problems.

Now compare this to corrupting potion. Corrupting potion gives 375 health return, more with the biscuit mastery. That means up until 5 minutes it has way better health return than Doran's shield. On top of that it gives a fairly strong DoT and and restores 225 mana. Most champions aren't going to go 5 minutes in lane without having stats matter. Most likely the weak attributes of Doran's Shield are going to cause whoever is using it to give first blood to the enemy. In addition to that, Doran's shield doesn't scale, so it will never be more useful than any of the Doran's items or the corrupting potion.

Then there's the 2nd passive, the damage reduction.. This is a cool concept that just sucks in practice. 8 damage reduction only from single target abilities and AAs is already fairly weak. But the big problem is that it factors in AFTER resistances. This means you could easily be getting less than 4 damage reduction. It's also a unique passive so there's really no way to stack this to actually make it slightly effective.

So we pretty much all know it's shit. Onto the fix.

If Riot doesn't want to rebalance the awful passives or make them non-unique, they just need to use the same formula they use with the other Doran's items. It would look something like:

Doran's Shield: 450 gold 80 HP + 10 AR + 100% HP/5. This would make it worth roughly 700G, still far less efficient than the ring and roughly equal to the blade. You could also lower then hp/5 to 75% and make it non-unique, similar to D blade and D ring.

Our other option is to change the price to 400 and lower hp/5 to 75%. This allows one additional potion which makes sense with D shield being purely defensive. It also makes it the same price as Doran's ring.

Either change would result in Doran's Shield being a practical starting item that can hold it's own with the other starting items. Remember that starting items are very gold efficient and strong, in exchange for having awful resale value (with the exception of cull) and taking up tons of slots if you buy more than one.