To overcome short key length of DES, Triple DES was devised. What would go wrong or why didn't people redesign the key scheduling algorithm to use longer keys?
Same as the reason of why not switching to AES (or any other 80+ key bits algorithms available at the time).
Because we want to reuse whatever "hardware" (chip design) resources available to do the blockcipher calculation, and re-designing the key schedule would prevent us from doing so since the existing hardware cannot compute the new key schedule.
The confidence that we have in the security of DES comes from years (decades) of cryptanalytic attempts and research. Any internal change would mean that we would have to re-analyze everything from scratch, and if we are doing that then we may as well just design a new block cipher that is better in many other ways. (For example: block size, speed, code size, etc.) Note that block cipher constructions are very subtle, and small changes can completely change their properties. Although this is more true about the S-boxes than about the key schedule, we would still need years to be confident again.
Just changing the key schedule would improve the security of DES only moderately (at least in the model where high data complexity below exhausting the codebook does not render an attack impractical). For instance, chapter 4.6 of the book Differential Cryptanalysis of the Data Encryption Standard by Biham and Shamir describes an attack on DES with independent subkeys that has a complexity of roughly 2^60 DES operations.