It isn't exactly a "3rd party" service... "SeedPicker" is simply
an (open-source) tool that basically bruteforces the 24th word for a BIP39 seed phrase when you give it the first 23 words...
Like it encourages you to
put 342 pieces of paper into a bag and "randomly" draw one out, then pick 1 of 6 words on the paper by rolling a die to get 23 words, then bruteforce the 24th using their tool
imo, it's still a terrible idea... 24 word seeds aren't chosen by selecting 23 words and then bruteforcing a 24th... they're created by generating 256 bits of Entropy and then encoding that using
the BIP39 algorithm and wordlist.
I have no idea what the entropy of pulling pieces of paper out of a bag is... but I'm not convinced it is going to be 256bits worth!!?!
So, I concur with the OP that you'd be much better of using another hardware wallet like a Trezor or Ledger or even an airgapped BIP39 compatible wallet (Electrum on a LiveOS etc) to create the "3rd" seed/key etc:
1) Instead of using seedpicker.net for the 3rd emergency key, wouldn't it be better to generate the key on another hardware wallet e.g a trezor? This would not be used to sign. The xpub would be imported into the multisig and the multisig would never know of the hardware wallet that generated that key. This would only be needed for recovery if 1 key was lost in a 2 of 3 for example.
As for this:
2) How many copies of each seed should be stored? Would there just be one for each seed so e.g for 2 of 3, there would be 3 in total, or would there be multiple of each? Maybe if that is a consideration, then a different quorum should be chosen.
It's not really dependent on the quorum really... in that m-of-n will always require (a minimum of) "m" seeds and "n-m" xpubs... it doesn't really affect how many copies of each seed should be stored.
So it really depends on your personal risk tolerance, I guess... for 2-of-3, as long as you have access to
at least some combination of 2 seeds and 1 xpub (where they are all from different "co-signers":
SeedA, SeedB, xpubC
SeedA, xpubB, SeedC
xpubA, SeedB, SeedC
Then you could easily recreate the wallet and sign transactions.