Yes, the computer browser service must be running if you want browsing by Netbios name to work (that is, the old computer browser system which came from Windows NT). This system is being phased out, basically because it simply will not run with IPv6.
If you are using the computer browser service, you also must have Netbios over TCP/IP enabled in the NIC properties. This also may not be enabled by default in modern versions of the server OS.
Incidentally the term PDC is also obsolete. That also comes from NT. These has not been a PDC since Windows 2000 came out. All DCs are equal (in theory at least).
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