All other ways will be just wastage if your sql agent services getting shutdown abruptly. Its good to find out reason for that and fixes that will be probably the best solution. Why don't you change the services parameter to start automatically when it stops.

But still you can try the other way around:- Query the msdb.log_shipping_secondary_databases and related table from other server (May be Primary) using link server and send alert.

Regards

Dhananjay