ppp-out default route disappears 1 second after interface is enabled

You still don’t need a periodically scheduled script to check the path transparency because you can use the scriptless failover as @Zacharias recommends.

But you may need an on-up script in /ppp profile to update the gateway of the lowest level recursive route via the wireless modem with the IP address of the remote end of the ppp link if it sometimes changes (different mobile operators behave in a different way).