The current possibility is that the NetworkManager is taking down the secondary interfaces and routes even though we are not using it to manage them. Restarting works because the network setup that we do happens again on boot. NetworkManager is the default system in RHEL/CentOS 7.
What is supposed to happen is that we set up the DHCP client to bring up the other interfaces (we write a dhclient config file for most distros) and then the DHCP client maintains the interface over time. So far this works correctly in other cases.
Running these commands and pay attention to the BOOTPROTO line:
$ cat /etc/sysconfig/network-scripts/ifcfg-eth1
$ cat /etc/sysconfig/network-scripts/ifcfg-eth0
As a workaround, updating ifcfg-eth1 to set BOOTPROTO=dhcp should fix the issue.
A permanent solution is being considered by our Network Engineers, but there is no date for going on production.