Wonderful thought Steve.
I guess if Organization is not serious about DR or they feel they can live with the downtime caused due to unavailability of DR then DBA's can't do much even if they intent to do so. Infrastructure is needed and has to be funded by the organization. DBA's are not going to fund it from their pocket.
But yes, proper backup strategy can be planned to be prepared to recover the data in case of any disaster. But even for that Infra is needed such as proper Network Bandwidth, Disk Space, Tapes etc. As a DBA we can make sure to have proper Backup strategy and periodic sanity of the backup files to make ourselves in the position to recover from any disaster but as we all know it needs more downtime.
Recently, in one of the Job Interview, I was asked a question to write down the query to fetch the latest database backup chain and backup and compressed size of the backups. I think we can schedule such kind of alert or make it part of periodic checklist.
I must stress DR should be there. If organization do not intent to spend much on DR then they can have the cold DR which will be made up only when there is disaster. It can be done using Logshipping, AlwaysOn on standard edition without paying additional cost on the SQL license.
I've recently found Microsoft Azure offers services for such kind of needs when one have pay only if there is actual use of the instance and not for replicating the data. Although, there is comparatively slight cost involved in setting up the instance. Resource utilization can be configured dynamically or it can be low on normal situation and at the time of disaster resources can be increased in seconds.