What should businesses start to automate first when starting off with an enterprise scheduling tool?

I would also start with jobs related to IT infrastructure like backups, cleanup jobs, etc as they are the easiest to setup and the most vital one's for your enterprise.
This helps you also to familiarize with the tool and get ideas. At this stage you should also set up the environments (dev,test,prod,...), the development standards and a naming convention. Afterwards its more difficult to change the habits.
Involve as much persons as possible at the beginning and offer them trainings in order to get them to use the tool.
Naturally I would than migrate whenever possible the system task related entries from existing tools like cron, Windows At, IBM iSeries Jobscde, etc
in order to have a central overview of all processes with its dependencies.
When it comes to the business processes, start with the core system (most important business application) and automate the existing processes.
Think also, which additional manual processes can be automated. Start there with the one's, which are important to responsible persons an which takes time for manual processing. It is most often used as showcase to prove it's real value (time and resource saving) and promote the tool across a company.
Finally you do the processes of the remaining (satellite) systems.

If we refer to the scheduling part of application servers there are 2 types of processes. Technical and business. Some may be dependent and others completely independent. For the technical part, you should provide the technical schedules necessary for the operating and security of your application environments. Scheduling backups (server and database), Scheduling tehnical process of the database (index rebuild, etc.) ...
For the business part, it is up to the business teams to express their needs in terms of automation according to their business department. (account, flow, reporting (datawhare), banking, financial, marketing, etc.) There is no order of prioritization from a scheduling point of view. It is up to the business to set its own priorities. But above all it is imperative to define the standards of coding objects. Define and install the platforms scheduling engine. (development, training, integration, preproduction, production), define the different calendars. And validate the shedules plans with the business departements on the different environments before switching to production. And first off all don't forget to contact the operating team to find out their requirements.

With my experience, the first step to automate your IT Production is back-up of your servers and common infrastrcture or exploitation tasks which can run the night. After that you can automate HR software and business specific application when your workers have got a good experience.

I think technical tasks and business application after.