Usually i see people buying elixirs when they already have 6 itens and want to spend the remaining gold. But in some games, i see players buying elixirs early in the game, around level 11 or 12.
Is there a strategy of buying these elixirs early?
Several points are to be considered for elixir use:
For reason number 4, you find that when a player is fully stuffed, he'll then go and buy elixir's with his remaining money. But that only happens on low / medium level games (never seen a pro game last this long).
However, if you look a high elo games, you can see player's use elixirs early or mid game. I will try explain, but the correct use comes with experience, team play and experience (because you need twice as much experience compared to team play :) )
From what I have seen, elixir helps a team who is late on gold secure or come back into a game with a team fight. If team takes elixir, it will compensate (and maybe more) the lack of stuff / level it has and allow her to win a team fight. However, enemy team seeing the elixirs being used will just retreat to avoid team fight. So it helps secure secondary objectives (drake, nash, clearing jungle creeps and wards to take map control, pushing a tower, etc.) in case enemy team avoids direct confrontation. Whatever the situation, you have to make sure when buying the elixir that the return on investment will be higher than the cost of the elixir + what the other team will have won as gold in the same amount of time.
Another use seen in the current world championships is to allow a player whose lane was being totally denied to him to get a buff allowing him to get some last hits and resist to the jungler's ganks.
To sum up, it helps you (or your team) catch up in power if you feel you're not up to the match, but you have to make sure investment is worth it. And this judgment comes with experience.
In regards to that particular elixer, It can be used like a heal. Stolen from wiki
The other elixers are as you said mainly used late game to help boost stats when no inventory slots are available.
In some case it can gives you a strong asset for the early game for example Ezreal jungle can use the Elixir of Fortitude for a fast jungle clear and nasty ganks by increasing his life pool and his damage (=> his Q) . With a good pull he easily kills blue without taking any hits.
As Fiddlesticks I sometimes take Elixir of Brilliance at level 5 just before the 2nd blue buff then I go drake at 7:30. With the CDR of the blue + elixir you can chain Drains and solo the drake without losing life and normally you will hit lvl 6 with the EXP; then you can ult bot for a double kill !!! :-D With the 20% CDR blue + 15% CDR Elixir it's an ultimate which will come back very fast.
As Tryndamere you can jungle pretty well with the Elixir of Agility boosting your crit chances and you attack speed will allow you to get your 6 really fast.