Yes, SHA-256 is safer than SHA-1 beyond the hash size.
In particular, SHA-256 truncated to 160 bits is significantly more collision resistant than SHA-1 is: we can make a collision for SHA-1 with effort like $2^{61.2}$ hashes, versus more than $2^{80}$ for an unbroken hash, including SHA-256.
When it's known that all targeted platforms are 64-bit, SHA-512 (and truncated versions) is often preferred to SHA-256, because it's faster for large input. It's probably also rather safer (but safer than safe is rather pointless).
SHA-3 and SHAKE have the further advantages of not having the length-extension property (contrary to non-truncated SHA-1, SHA-256, SHA-512), and (for SHAKE) of having customizable output length.
As far as I understand, the hash is used as a password generator, I imagine from a master secret, a site name, and a login. In theory, we'd need a MAC with the master secret as key, and the rest as the other input; or a hash designed to be a MAC simply by starting the hashed message with the key. SHA-3 is superior in that regard. If we use SHA-1 or SHA-256 we should use HMAC-SHA-1 or HMAC-SHA-256; but OTOH there is no known attack if we do not.