I think it's good that the more complex Summoner Spells are for later levels. New players get to learn what each Summoner Spell does and they don't get overwhelmed with a bunch at once.
Ghost and Heal are untargeted abilities, the easiest abilities to use. (Heal does have a targeted component but it can be used without a target.) You get them at level 1.
Barrier is another untargeted ability and you get it at level 4.
Exhaust is a targeted ability and it doesn't have easy to understand effects for a new player so I think it would be better if its level requirement and Ignite's were swapped. (A movespeed slow is somewhat easy to understand, but a new player would likely have exceeding amounts of difficulty using Exhaust to reduce someone's damage output.)
Cleanse is another untargeted ability but its effects are less easy to understand and it is difficult to use effectively, so it's understandable that it is unlocked at level 6. I honestly think level 8 is more appropriate but I get why Flash is unlocked all on its own without any other accompanying Summoner Spells - Riot wants players to focus on Flash.
Teleport is a targeted ability but it is pretty easy to use and easy to see why it is so powerful. Level 6 is understandable.
Flash is a skillshot and it is really versatile. It is easy to use but I don't think most new players would be able to appreciate its effects.
Ignite's level 10 requirement is something I don't understand. I feel like this is a case where a spell is level locked because it is a cool spell and Riot wants new players to try and unlock this cool spell. It is an easy to use, easy to understand targeted spell.
Smite's level 10 requirement is completely understandable. It is very difficult to jungle without many mastery points and rune slots, and most new players wouldn't understand how to jungle anyways. It is a much more difficult role to get into than laning.
Now, as for the unfairness aspect, yes it is a bit unfair, but since everyone is new, players won't be able to use the Summoner Spells to their full capabilities (for example, I got executed three times when I tried to jungle as Rek'Sai for the first time despite being instructed by my friend and being Rek'Sai, who was pretty broken at the time I was playing her). Furthermore, players level up quite quickly in the early levels so the spells will not be locked for long at all.