At what level is too much change for an organization to absorb? Modernizing a 30-year-old mainframe system to the cloud  + Modernizing a new CX framework with the introduction of omnichannel capabilities +  Modernizing a new data platform and moving away from the hub and spoke data warehouse to a data lake on the cloud +  Several other mid-level enhancements... Is it too much to take on at the same time? Or it all depends... How do you measure the appropriate level of change to introduce to an organization (Both IT and business)?

To add - make "the business" an active participant in the IT game - they own the context.  Remember it is their goals you are enabling.  They must decide...

Remember; The "business needs IT and IT enables the business."

An organisation can absorb lots of change - its how you "chuck up" the change which is important and prioritise.  Breaking it up.

Put the decisions to the board / C-Suite in simple decisions they can make quickly.

You mention Mainframe - so lots of bulk processing etc.
1 - So with Mainframe think about each component of it as a service rather than a technology - what does it do?  What is your context? What is the goal?  Migrate Services or/and Migrate and Transform.  
Business language only
2 - think about the concepts of it - what and why.  The conceptual capabilities you need to enable the goals
3 - Solution Service - which consume capabilities
4 - and finally technologies that enable / risk manage - technology enables the capabilities and thus services your need to achieve your organisational goals

Don't think about technology - think about little pieces that make a Service.

So the Org can absorb lots of change - but it needs prioritisation as it can't be all done at once.  Focus on key deliverables and outcomes enabled by technology

It is not how much change but how much change over time.  I find large changes are best if they are phased.  I would encourage that you engage your user community and stakeholder in Organization Change Management (OCM) sessions to discuss the object and establish an orderly process and acceptable schedule that is acceptable not only to the user community, but to the staff who have to support and maintain the system.