Does the NTAG216 scan better than the NTAG213?

No. The only technical difference between the NTAG216 and the NTAG213 is the amount of available memory for NDEF (URL, text, etc) storage. The NTAG216 has 888 bytes compared to the NTAG213 which has 144 bytes.

However, if you are storing a weblink/URL, then we strongly recommend using the NTAG213 chips. The only reason to use the larger memory tags is if you are directly storing large amounts of data onto the tag (and note that storing vCards onto NFC tags doesn't work!).

The NTAG216 and NTAG215 are _not_ better chips. In some cases, they may even perform worse if the reader decides to download the entire memory before taking action.
Seritag - 11 Jan 2023

I tried an NTAG216 and I thought it scanned worse. The NTAG215 seemed OK but the NTAG213 seemed to respond quicker than the 216.
Reader - 13 Jan 2023