Everything you say is true. The reason I had a problem at one time was probably because my router was in decay and later failed but I don't really know that for sure. There may have been some other problem as well. When I replaced my router the DHCP function worked as expected. One thing I noticed with my new Netgear router was that its selection of IP addresses with the DHCP function did not stabilize for several days. As a result the various computers on the net had changing IP addresses every time they were booted. I talked to the Netgear tech support person and he had no answers.

The changing IP addresses posed a problem for inter-computer communications as each had to know the IP address of the others. I later found a feature in the Netgear administrative set called LAN IP Setup which allowed the internal IP addresses to be fixed once they had been selected by the DHCP function.