A method and system for controlling operation of a first access node that supports operation according to a first radio access technology (RAT) but does not support dual-connectivity operation according to the first RAT and a second RAT. A controller detects a high extent of occurrences of dual-connectivity-capable user equipment devices (UEs) being connected with the first access node when the dual-connectivity-capable UEs could instead connect with a second access node that supports the dual-connectivity operation. And in response, the controller suppresses coverage of the first access node, such as by reducing a maximum signal delay that the first access node applies for determining whether to accept random-access requests from UEs and/or (ii) reducing reference-signal transmission power of the first access node.