I think mainly because "hybrid" is a choice in item build path, not a core champion class.
There are 3 basic archtypes in League with subtypes (arguably 2).
AD - marksman, fighter, assassin
AP - mage, assassin
Tank - AP or AD based
With the 3 archtypes, the doran's items as of current can be used in each of these molds.
If you look at the items there is a valuation on basic stats that you have to consider and if you try to diversify too much on a basic item, you won't have enough valuation to get anything significant for a "hybrid" build.
Valuation stats:
Health: 200g for 150 HP = 1.33g per HP
AD: 360g for 10 AD = 36g per AD
AP: 435g for 20 AP = 21.75g per AP
Lifesteal: 440g for 8% = 55g per 1% lifesteal
Attack Speed: 450g for 15% AS = 30g per %AS
MP5: 180g per 25% base regen = 7.2g per 1% base regen
HP5: 180g per 50% base regen = 3.6g per 1% base regen
Note: HP regen value is very hard to quantify as every champion's base is vastly different with the lowest being 2.5 and the highest being 11.005 per second. I would say the vast majority fit between 6-8 HP per second, so for sake of argument (and easy numbers for comparison on Doran's Shield) 6 HP per second will be considered the average baseline.
Doran's blade (440g): 93.1g value from HP, 252g from AD, 165g from Lifesteal = 510g value
Daron's ring (400g): 79.8g from HP, 326.25g from AP, 33-50% effective base mana regen (call it 237.6g for sake of argument) = 643.65g
Doran's Shield (440g) = 106.4g from HP, 360g from HP regen, (no comparison for passive) = 466.4g
Considering that all starting items allow for 1-2 health pots and typical starting gold is 475g, a starting item would have to fit this kind of valuation. I am going to assume for a moment that Doran's Ring mana regen valuation is off, and let's figure the typical valuation should be around 500g.
Now let's evaluate your suggested item:
Doran's Cap: 66.5g from HP, 144g from AD, 195.75g from AP, and 240g from AS = 646.25g valuation.
Prospective changes to get hybrid stats and proper level 1 valuation:
Doran's Cap: 55 HP, 5 AP, 5 AD, 5% Attack speed = 512.0833g
This also has the added flavor of having all stats be 5, although that wasn't really intentional when I was first running the numbers to make them fit.
On another note, there are already some other hats in the game (it might be strange to have a Deathcap and Doran's cap) so might I suggest something that resonates more with the stats with respect to item location such as Doran's Amulet?