The Cisco ISR 4321 router is not giving Internet connections to the LAN PCs when the WAN interface has a /30 mask length (mask of 255.255.255.252), which is the actual IP address range I got from the ISP for my leased line. It has no problem using a /24 mask length (mask of 255.255.255.0).
Why is this happening?
It sounds like even though that's your block assigned to use the ISP hasn't segmented it out so the ARP broadcasts are working improperly to resolve, this is usually done by smaller ISP in order to conserve IP space. If they were to give an actual /30 to every customer they would be losing 3/4 of the IP addresses they have to allocate (their router's IP, network address, broadcast address). I would confirm that with them, but I've seen it before. Otherwise if that's not the case you can sterilize your configuration and post it so unique problems can be dissected.
/31 for the connections; they all seem afraid of it. The RFC for using /31 has been around for 15 years! That's several generations in Internet time. Using /31 would give no waste for IPv4 address blocks. They also want to use /126 for IPv6, but that RFC has been obsolete for years, and the current RFC says to use /127. Granted, IPv6 doesn't have the address shortage that IPv4 has, but they should actually use what the RFCs say is correct.
/31 doesn't work. If I did, it would be rejected as non-compliant with IETF standards. As far as IPv6, the /126 was rejected because of the "Ping Pong" attack, and using /127 makes a link immune to that particular problem.