What areas in the finance team do you think are critical for a CIO to be more involved in?

The CIO needs to have trusted relationship with CFO.  Next, the CIO needs to have strategic relationships with those the manage the OPEX/CAPEX and with Sourcing/Procurement.  A CIO that doesn't manage money as their top priority will no capability to deliver on company needs/ambitions AND employee experience.

Budget oversight, CapEx strategy, cost distribution, and revenue operations

A connection with the CFO is critical. Without alignment and support it's hard to get buy in on how money is spent.  Also, a relationship with the financial analyst (s) is key to ensure the CIO understands and has influence over the pockets of IT money being spent. Especially at budgeting and strategic reviews.   

Both thoughts already expressed align with my thinking as well.

I would only add that understanding the data flows that ultimately produce the financial statements are also critical.  Auditors are looking to ensure that IT controls are sufficient to support all financial transactions and the final result of those transactions is the monthly/quarterly/annual financial reports.

Since taking over ERP, I've been involved in the following: 
1. Integration of ERP with Financial Consolidation  
2. Integration of ERP with FP&A 
3. As Group CIO, the need the streamline and differentiate ERP finance functions across very different business units / lines of business and Group vs BU demarcations. 
4. HR, Finance and IT integration