Fish can't live without water, as/which is known to all.
這裏是（非限制性定語從句），而 as/which 替代這個句子作主語，所以 as/which 是不可省略的。假如 as/which 省略了，變成您寫的模式，句子就變成了斷句了，因為：
Fish can't live without water is known to all. 這壹句的主語是 fish，但有兩個動詞，壹個是 can't live，壹個是 is，兩個動詞之間沒有任何的連詞，就變成了斷句了，所以需要使用逗號，變成了非限制性定語從句。
As is known to all, fish can't live without water. 其實 as 本身並不是（關系代詞），因為關系代詞只有5個。假如您查考牛津詞典，as 這個單詞本身的詞性只有：介詞、連詞、副詞，但沒有（代詞）這個詞性的，也就不是（關系代詞）了。所以語法書基本上都把 as is known to all 劃入到（固定結構）中了。
Fish can't live without water, which is know to all.（這個應該是按照現代語法完整的原句）。但當非限制性定語從句提到句首時，是無法用 which 引導的，所以把 which 改成了 as，也就是把 as 的（連詞）詞性視為（關系代詞）賦予 as 有 which 的功能。這個是現代語法的解釋。as 在比如和 such，the same 連用時，也是（連詞）詞性視為（關系代詞），比如：He is such a man as I honor. 所以查考語法書時，您會發現，語法書通常使用（視為）關系代詞，而不會直接說 as 就是（關系代詞）的壹個原因了。
類似於這樣的（固定結構）還有比如：
as is said above
as is often the case
as is reported
as has been pointed out
as has been said