How to configure vlan for monitoring on l3 switches?

Question

Network routing

Good day everyone!
There are Extreme Networks L3 switches. Enabled routing between VLANs (enable ipforwarding). In these switches, ports can be tagged (similar to Trunk) and untagged (similar to Access).
Imagine a (spherical in vacuum) structure that consists of a main switch (MC), two secondary switches (K1, K2), a workstation (RS) and a network printer (P). Port 1 of the GC is connected to port 1 of K1, port 2 of K1 is connected to the workstation. Port 2 of the GC is connected to port 1 of K2, port 2 of K2 is connected to a network printer:

ГК(1) - (1)К1(2) - РС
ГК(2) - (1)К2(2) - П

Workstation IP address 192.168.1.2/24, gateway 192.168.1.1.
Network printer IP address 192.168.2.2/24, gateway 192.168.2.1.
There are two VLANs on the GC: vlan 10 "Hosts" (port 1 tagged) with IP address 192.168.1.1 and vlan 20 "Printers" (port 2 tagged) with IP address 192.168.2.1.
K1 has one VLAN: vlan 10 "Hosts" (port 1 tagged, port 2 untagged) without an IP address.
K2 has one VLAN: vlan 20 "Printers" (port 1 tagged, port 2 untagged) without an IP address.
Once configured, the workstation sees the printer, and the printer sees the workstation. Everything is great. It was a hint.
The problem is the following. Let's say I want to monitor the availability of switches (ping) from a workstation. To do this, I create a separate VLAN 30 "Control" on the GK, K1 and K2 and assign these VLANs IP addresses 192.168.3.10/24, 192.168.3.11/24 and 192.168.3.12/24, respectively. On the GK, I add ports 1 and 2 to vlan 30 as tagged, on K1 and K2 I add port 1 of each switch to vlan 30 as tagged. After that, from the workstation I see only the GC (192.168.3.10), but I do not see K1 (192.168.3.11) and K2 (192.168.3.12).
Why?

Answer 1

On switches K1 and K2, you need to specify the default route 192.168.10.10:
configure iproute add default 192.168.10.10

Answer 2

I don’t know much about Extreme Networks specifically, but do you let VLAN30 trunks through?
If strictly as you described, then no. It turns out that VLAN10 goes to the PC from the PC, then it "routes" and should go back via VLAN30, but you don't have VLAN30 in the trunk.