Does having your domain and web host at the same company reduce DNS propagation times?

Propagation times are controlled by the time to live (TTL) that is set on the DNS record. The only way to get DNS changes live faster is to have a shorter TTL specified.

GoDaddy may be offering to set a short TTL for you, but there is no reason that a short TTL can't be specified at most DNS hosts.

Default TTL values vary widely between DNS hosts. Some may use 30 minute TTLs, but most set 12 hour, 24 hour, 48 hour, or even 36 hour TTLs.

Short TTLs may not even be honored by many caching DNS servers and they may cache records longer than indicated when they deem TTL values to small. Regardless of what your TTL is set to, you should plan to have you old server running for three days after you switch the DNS records. 95% of your traffic should move over within the TTL expiry, but a small amount may hit the old server for days.