However, if it would have gone on this path, I'd expect some of other exchanges' customers have the same problem - at least those with no 2FA set.
They do, just not all at the same time like this, since this attack involved both a leak of data and a security exploit in Coinbase's SMS systems. Exchange accounts get hacked all the time, especially if they are using no 2FA or weak 2FA like SMS.
Some still keep an awfully lot of useless mails and sensitive data in their mailboxes, but scanning so many mailboxes to find out whether they're Coinbase customers or not may not be a small job (of course, it can be automated for some of the servers).
It would be pretty trivial to write a bot which would log in to every Gmail account (for example) you had credentials for and then run some quick searches for "Coinbase", "Binance", "Bitfinex", etc. Also, the email and password leaks could have come from a crypto related service. If some faucet, ICO, bounty, etc., leaks or sells a database of 10,000 users, then you can be certain that the vast majority of them will have an exchange account, and knowing the kind of users who sign up for bounties and airdrops, probably a far higher than average percentage of them reuse the same password across all their accounts.