generic host monitoring in GTM

If your requirement is - "The major objective is once one host failed and then only respond the DNS query using another available host."

This means the second member in the pool should not be taking traffic unless the primary member goes down. This is achieved with Global Availability load balancing method.

Round Robin distributes traffic to both the members.

When you say - "it looks GTM cannot monitor the generic hosts (pingable/reachable from the GTM tmsh)."

Are you stating the members are always marked down, what is the monitor configured for the members.

Can you try something with below and see how it works,

Creating servers
create gtm server 10.10.10.10 datacenter "USA" monitor tcp_half_open addresses add { 10.10.10.10 } product generic-host virtual-servers add { 10_10_10_10_80 { destination 10.10.10.10:80 monitor tcp_half_open } }
create gtm server 20.20.20.20 datacenter "USA" monitor tcp_half_open addresses add { 20.20.20.20 } product generic-host virtual-servers add { 20_20_20_20_80 { destination 20.20.20.20:80 monitor tcp_half_open } }

Creating pool
create gtm pool gtm_pool load-balancing-mode global-availability verify-member-availability enabled ttl 30 members add { 10.10.10.10:10_10_10_10_80 { order 0 } 20_20_20_20_80 { order 1} }

Creating new wideip - testing.wideip.us.com
create gtm wideip testing.wideip.us.com ipv6-no-error-response enabled pools add { gtm_pool }