Do you have by chance some maths/probability calculation for one attacker attempting to hit a 12 words seed for Electrum wallet? I guess the possibilities are higher than using random private key of 32 characters as brute force can be done using a dictionary?
A bitcoin private key of 256 bits in size provides 128 bits of security. (See page 4 of
https://www.secg.org/sec2-v2.pdf)
In seed phrases, each word encodes 11 bits of data. A 12 word Electrum seed phrase therefore encodes 132 bits of entropy, and so is therefore slightly more secure than a random private key. A BIP39 12 word seed phrase also includes 4 bits of checksum, and so encodes 128 bits of entropy, the same strength as a random private key. BIP39 seed phrases can also be longer than 12 words, and each additional 3 words provides an additional 32 bits of entropy (with 1 more bit of checksum being added to the end). The longest BIP39 seed phrases at 24 words therefore provide 256 bits of entropy.
Private keys in hexadecimal are 64 characters, not 32.
The strength of 2
128 is 3.4*10
38.
The strength of choosing 12 random words (in the case of Electrum) from a list of 2048 words is 2048
12 = 5.4*10
39.
So even the shortest seed phrases are at least as secure as individual private keys.
