But it would be nice if important details like the derivation path was somehow possible to be encoded into the seed phrase.
As you point out, Electrum seed phrases do this. Basically, when Electrum generates a seed phrase, it then hashes it and checks if the hash starts with the correct version number. If not, it increments the entropy by 1 and tries again, until it reaches a seed phrase whose hash does start with the correct version number. That version number tells Electrum which script type and derivation path to use, which is why Electrum seed phrases are either legacy or segwit and will only ever recover one wallet, as opposed to BIP39 seed phrases which can use any script type at any derivation path and restore a near infinite number of wallets.
they need to store the derivation path along with it
A better option is for the majority of users to just stick to the BIP44/49/84 standards and not mess around with custom derivation paths unless you really understand what you are doing.
no i would not want it doing that. but what if i entered something that wasn't my address and it actually passed the checksum? hopefully the probability of that is on the order of 1 in trillions or even more.
For legacy addresses, the chance of an incorrect address with the correct checksum is 1 in 4,294,967,296. For segwit addresses, the checksum is guaranteed to detect any error effecting up to 4 characters, and less than 1 in a billion chance of failing to detect more than that. So not quite 1 in trillions, but still incredibly safe.