Shouldnt't the alternate be far enough not to be influenced by the same weather system?
No, unless the destination and alternate are both below planning minima. For example, alternate fo EGLL is EGKK, but both are "fogged out" in a high preasure system, then you could end up using EHAM (for example) instead of EGKK because the wx is ok.
Note: EHAM may still be part of the same system (high pressure), but because the forecast is ok it's usable.