This sentence just doesn't seem natural without «мне».
«Мне так не кажется» is used for refuting someone else's point of view. 'So' / «так» refers to someone else's assumtions. So, if you're refuting someone else's point of view, it means it's still held by someone. So, you can't say «так не кажется», because for the people whose assumtion you're refuting, it does seem so.
In general, «не кажется» doesn't work well without dative. It's either «мне так не кажется», or «кажется, не так». I.e. if you're making generic sentences, you need to negate the assumption and not the verb казаться.
It doesn’t work well with «так».
«Кажется» is often used as a parenthetical word (similar to «наверное»), it’s not treated as a full sentence, it’s a parenthetical word to comment on the plausibility of some other sentence.
But when you use «так», you’re creating a full sentence. And as a full sentence, «так [не] кажется» sounds quite unnatural, unlike «мне так [не] кажется».
It could mean ‘It seems differently to me’, but it doesn’t sound too natural. There are better ways to say this.