Points on rental loan -- for short term rental (schedule E)

It depends, we may need more clarification. I performed the test with $1,000 cost and 30 years, using May 15, as the start date and the amortization was correct at $22 for a partial year.

You have the correct information and the steps necessary to achieve your amortization. If it's truly a short term rental and it is not used full time for rental activity, then it's possible the amortization could calculate low to zero.  

1. Was it converted from personal use to rental use?
2. Was it used for both personal and business use with the short term rental?
3. What were the number of days the property was rented?  

Steps Used in the test above:

1. Sign into your TurboTax Account > Search (upper right) > Type rentals > Press enter > Click on the Jump to .... Link
2. Edit beside the rental you want to work on > Scroll to Assets > Edit
3. Select Yes to go directly to the asset summary >
4. Add or Edit the asset for the closing costs and or points > Select Intangibles, Other Property > Continue
5. Select Amortizable intangibles > Continue > Enter the details about your costs including the refinance date
6. Continue > Select the Code Section 163 for loan fees > Enter the useful life (number of months of the loan)
7. The final screen will show the deduction.  

If you are able to provide more clarity we are here to help you further.

On what specific date did you acquire the property? Or if converted, what specific date did you convert the property to rental?
Depending on the dates and numbers, if acquired/converted in late December, it's perfectly possible to have $0 deducted for that first tax year.

Described asset - added cost and closing date of loan

You probably are aware of this. But I state it solely for clarity. Your closing costs also includes any pre-paid interest that was paid *at the closing*.  This amount may or may not be included on the 1098 if you get a 1098 for the tax year you closed on the purchase. You need to be aware and check that, so that you're not double-dipping on the interest deduction.