I still see a lot of confusion, let me help you out with this.
Let’s start from connecting F5 to your network. You have one internal VLAN (10.1.0.0) and one external VLAN, correct? And you mentioned that the VLANs are untagged, so you have 2 physical interfaces as well, correct?
For every network segment where F5 talks, it needs a self-ip address. This is pretty much to let your gateway and your LAN know you exist and to provide a MAC address for layer 2 operation. 2 VLANS means 2 self IP addresses in this case.
If you want the self-ip’s to respond to ICMP probes (ping), then you need to tune the self-ip lockdown behavior.
Last thing you need is a default route. In every deployment I ran, this is configured to point to the Gateway of the external VLAN so that packets destined to my client IP addresses are routed back where they came from.
Now, let’s configure a virtual server. You create nodes, you configure a load balancing pool, and you make sure F5 can talk with them (you already mentioned monitor’s green, so great!)
Next step is to set up the VIP responder. It can have any IP address and it’s not restricted to your VLANs! Altough it’s common practice to assignn an IP in the same network as your external interface, just because it’s easier to route. But don’t worry - if you want a public IP to be on F5 you can totally do it withous having to use public addressing on the connected LANs.
When F5 receives packets for that specific IP address AND port AND on the external VLAN since you configured such interaction (it has to fully match the virtual server responder configuration), it will accept the packet. There’s built-in counters in the GUI that help you understand if this is happening.
Now the F5 makes a load balancing decision and forwards the packet to the servers.
You said that from pcap files you see this is happening, and this is great, but .. do you also see response from servers?
My bet is that the server is trying to send the packet back to the original source, instead of routing it back to F5. An easy way to force this interaction is changing NAT configuration on the Virtual Server to “auto map”. This means that when the packet leaves F5 on the server-side connection, F5 changes the source IP address to one of his own self-ip’s (specifically, the one of the interface that was used to route the packet out).