You could have slightly messed up with the info. As far as I have understood multisig addresses, let me try to explain it in a simpler way.
An ordinary bitcoin address consist of only
one private key. In simpler terms, a private key is hashed to create a public key which in turn is hashed to create a bitcoin address. Whereas a multisig address has a collection of private keys. Let us consider that 15 private keys create 15 public keys. These 15 public keys are used to create a locking script. This script which consists of 15 public keys is hashed to create the multisig address. Based on the M of N which you have specified you need 1-15 signatures to unlock the script and spend the bitcoins.
A locking script looks similar to this
15 [pubkey 1][pubkey 2][pubkey 3]...[pubkey 15] 15 OP_CHECKMULTISIG
An unlocking script to spend the bitcoins present in the multisig address will be like
OP_0 [sig 1][sig 2][sig 3]...[sig 15]
Hence, to unlock the script we need to have an unlocking script which consists of 15 signatures from 15 private keys. Hence, spending the coins require the correct unlocking script to be provided for validation which can only be fulfilled by these 15 private keys. There is no other 16th key involved as you think.
Is it therefore much harder to bruteforce if quantum computers descend in the next few years?
Yes, based on my assumption. Bitcoin is designed in such a way that it is impossible to derive the private key from the normal address unless it is exposed. Even after the involvement of quantum computers, it would be not very difficult to find a single private key. But in order to spend the coins from a multisig address, 15 private keys needs to be involved which would be quite difficult as all the 15 keys are independent to each other.
I'm thinking of printing out 15 seeds and storing those seeds on a flash drive and putting that flash drive in a secure location
Bad idea. If the flash drive is stolen, all your coins would be lost as all the keys to release the coins are exposed to the hacker or the thief. Better to store the 15 seeds in 15 different locations so that it would be impossible to steal your coins.
is there some sort of private key that can be calculated out of all the other private keys that exist which relates to a "regular" address' private key?
No, I don't think so there is another 16th private key which can be derived from the 15 keys.