TurboTax says my child is required to file a return if their unearned income is >$1,250 but the IRS says >$2,500 (https://www.irs.gov/taxtopics/tc553). What am I missing?

The website is very misleading 

for unearned income less than $1250 is tax free  - at least for federal purposes

between $1251 and $2500 the income may be subject to tax at the child's rate 

above $2500 it's the parent's rate.

if all the child had was interest and  dividend income and capital gain distributions from mutual funds/etfs.

you can either 1) report this income on your return using form 8814. the child does not file.

2) or file a 1040 return for the child and include form 8615

potential disadvantages of 8814

1) no increased standard deduction if blind

2) itemized deductions are lost

3) early withdrawal penalty not allowed

4) parents may pay more in Net Investment Income Tax

5) tax on child 

6) the excess between $1251 and $2500 is taxed at a flat 10% rate. if these were qualified dividends and the child filed their return  the tax could be zero so using for 8814 can result in more taxes on the child's income