Congratulations!
Agreed.
Nope. DW already (wisely!) indicated that life insurance isn't a good idea. But if you want to commit to 20 years of payments, great! Pay your kid directly! That is, pick one or a couple low cost index funds (in a low cost Regular Savings Plan for example), and do that for 20+ years.
From age 15 days onward you can purchase an Integrated Shield plan for a healthy newborn. However, it doesn't have to be a private hospital plan. For anything truly serious you're surely going to head straight to KK Hospital, the best pediatric hospital we've got (and most likely the best in Asia). You can buy an affordable public hospital A ward Integrated Shield plan (and rider) that'll provide "run of the house" coverage at KKH, and that'll work great.
Since
@DW asked specifically about CI/ECI I assume the concern is about cancer as a notable (the?) example — childhood leukemia, let's suppose. Thus I think it's prudent to compare Integrated Shield plans (and riders) in part on how much cancer coverage they provide. Ideally that's a high annual number with no monthly sublimits. Last I checked Prudential PRUShield Plus ticks a lot of boxes, but YMMV. (One downside with Prudential is they don't offer a B1 "downgrade" plan except the Standard Plan.)
Aside from an Integrated Shield plan (an affordable one), I can't really think of anything. When you have acute medical care well covered I don't think you really need the CI/ECI stuff unless you already don't have emergency reserve savings. (Fix that problem if you have it.) I wish there were a way to buy a long-term disability insurance policy for kids, for example something like an "early" DII policy that has a 2 ADL disability definition. Of all the parental worries I have, long-term care for a disabled child is a genuine worry. Unfortunately I'm not aware of any insurance product like that, and it could be because the private insurance market can't figure out how to make that sort of product work.
Salespeople sometimes argue that kids are prone to accidents, so...buy PA insurance. No thanks. If there's an accident, you take them...to KKH. (Integrated Shield again to handle any big bills.) Then what? A S$10K PA payout (for example) isn't life changing money. And a PA policy doesn't pay anything if there's no claimable event.