No it shouldn’t. Move history has to be encoded so draw by threefold / fivefold repetition and fifty moves / seventy five moves can be claimed / detected.
The half-move counter in the FEN takes care of the 50/75 move rule. You do need the history if you want to implement the threefold/fivefold repetition rule, but it's rare that it is actually relevant for e.g. a chess engine's strength.
The move order matters, for detecting threefold or fivefold repetition, or the 50 move draw limit.
To adhere to that properly, you need to somehow represent all previous positions that could be reached again, and the number of times it has occurred. Of course you can get that by including all the move history, but it's also possible to prune it a lot, like any capture or pawn move can flush the history since no previous position is reachable. But it's still a bit more complicated than just representing the current position.
FEN doesn't account for this, deliberately leaving the history out of scope. It's a matter of preference whether you'd want a tool like this to handle those cases.
Chrome can supposedly support 32779 characters in the address bar[0], and a legal chess game should not exceed more than ~5900 moves, due to the 50 move rule. That will be enough to encode any valid game if you don't need to support IE.
Interesting. According to that stackoverflow, Chrome is one of the shortest URL lengths of the modern browsers (firefox/safari/chrome). However, it also notes you have to take CDNs and web servers and search engines into consideration, so the ginormous length limits of safari and firefox probably won't be useful any time soon.
very cool, made it all the way to a checkmate