It's just how い-adjectives work. They always have an い at the end, and especially ～しい is a common way to end an い-adjective.
PS: asking "why does this work this way, wasn't it easier to do it different" is in general something you can ask, but not get an answer. Languages aren't made by a few people who try to make it as simple as possible. And changes aren't made either, because the natives have learnt it one way, and they're not going to adapt their native language just because some foreigners have trouble learning it. It's like saying you're going to standardize spelling in a logical way in English. No one is going to use it, just because they're used to writing things one way, and they're not going to change it just for learners. The same goes for Japanese.
Well, some words, like in English, have silent letters. What would be the point of that? Japanese has particular uses.
The い is there for a few reasons. They pronounce it that way.
Why do they pronounce it that way? Because it's an い-adjective.
Why is it an い-adjective? Because it has the い in it! They speak it.
It probably doesn't have to be like that, but it just does. Because it's there, it indicates it's an い -adjective. If it doesn't, it's not an い-adjective. It'd have to be some type of な-adjective or something.
Why is it an い-adjective? First off, the い in い-adjectives is not to be thought of as being totally superfluous. You especially shouldn't be dismissing it just because it has し, as in the "i" vowel that exists in 's h i'. It's nothing to do with the fact that vowel is there. It's not because it's a い-行 character for it to count. It's about the character い. Only い.
Why only い? The い itself is actually what is basically a copula. It's special. It's combined with an adjective and just became part of the word. How that came to be, I won't really go into it here, but this is just how it's formed. It's fixed that way. For basic use, all you need to be aware of is that this is the type of adjective that has an い in it. There are a couple exceptions, belonging to な-adjectives, which are the only ones that are classed there for some reason. い-adjectives exist, because they're the original adjectives used in Japanese. な-adjectives are mainly contemporary adjectives, which is also part of why they work out like nouns, since some of them were nouns as they were introduced to Japanese in later years.
Consider this: 同じ. Sure, there's an "i" vowel. But, no い! There's a clear difference. This wouldn't make it an い-adjective, so simply it's not one. It must be treated as a な-adjective. Simple. It's clearly nothing to do with the fact there's an "i" vowel you would hear as if it was an い-行 character. It just simply and explicitly has to have an い to be an い-adjective.
One other thing to note: There aren't any words that go the other way around to look like they should be な-adjectives being actually い-adjectives. So, it's the fact there's pretty much only two words, きれい and 嫌い, to watch out for. Basically, you can consider this guaranteed that one of the differences is to see whether there's an い in it or not, all apart from those two words. There's just only a special case consisting of words that are い-adjectives, and this is related to いい. It just has to conjugate differently, by deconstructing it to it's old form, よい (良い). It's because of the fact that it has the い in it, that shows it is still an い-adjective. Once you have よい (良い), conjugate like any other い-adjective. Simple as.
