If it doesn't exist on exchanges it should still work fine with the internal market. Even if your goal is to buy HDB using funds from an exchange or external fiat, you can buy HIVE and then trade that for HDB using the internal market. There really only needs to be one path to external value for it to work, the rest is gravy.
As for Korean pumps, I see it as a stress test. The stronger the stabilization mechanism, the less successful pumps will be and that in turn reduces the appeal of trying to pump it at all.
One should not expect perfect stability though. Even the granddaddy asset of them all, USD, has value fluctuations due to market disruptions and speculation, sudden large ones at times. That's inevitable. What is important is that relative stability is maintained.
Would it be better if it was a platform only token, rather than on exchanges at all? I have no idea if it can be taken off exchanges.
Also, wouldn't it work better to have a second layer token act as the stable coin, instead of one integrated into the blockchain itself?
Could be advantages to a second layer token, but also disadvantages. One being HDB is extremely simple at the blockchain level. The risks of errors in complicated smart contracts are significant in practice. I'd be more comfortable holding something like HBD than any complicated second layer token that isn't extremely well audited and proven out (DAI, itself, is probably okay, but many others trying to do this are likely not).
I don't see any advantage to not being on exchanges, nor can you prevent it. Just have to deal with it.