[CentOS] default gateway outside of the LAN

[James Hogarth]

> 10.1.1.1 via 192.168.10.1 dev eth0
> default 10.1.1.1 dev eth0
>

You need to revisit your networking knowledge ... specifically what a
gateway is...

You need a router (gateway) on 192.168.10.0/24 (presumably 192.168.10.1 in
your example). On this you have a default route to the next hop of
10.1.1.1...

In terms of routing systems on 192.168.10.0/24 have no knowledge of
10.1.1.1 and no ability to influence the route other than the next hop on
the same local link network as them (192.168.10.1 in this example).