How can you balance the needs of different business units in your strategic plan?

This is great, but is almost better done once the Vision has been decided. Without the Vision and supporting metrics (the real KPIs that help define the vision and help you measure others towards it), you cannot see how well different units' performance metrics align with it. Ideally, you should understand the high level state and performance of your business units at all times (otherwise, what are the Board actually doing?!) and have this to have when looking at your new vision.

Don't forget to include metrics here. A vision is great but one that is vague and woolly with no actual targets will not guide anything. Remember that the purpose of the Vision is to guide the direction you want to take the organisation and help you make informed decisions about what to do, i.e. things that get you close to the Vision.

Although valid, I would first decide on my interests and then look at various concerns that could impact my abilities to obtain certain objectives. A clear set of goals/objectives is a key in any successful strategy.

Defining Missions, or change journeys, is a useful approach here. By now, you should have your vision and targets against measurable KPIs, and your current state backed by similar metrics. Your next step is to compare them and see what changes you need to make. These should be in terms of outcomes rather than outputs - "decrease customer churn from 40% to 10%", say, rather than any specific measure. These missions give you a solid basis for challenging your business units to come up with specific objectives, or things they can do to improve those metrics. As a board, your job is now to encourage the units to collaborate effectively, and review, prioritise and fund the objectives that will make the highest value contribution to the Vision.

A business strategy needs to be board led, but it cannot be only delivered by leadership. When leadership do attempt to do everything themselves, they ignore they large pool of knowledge, experience and skill in the organisation, risk making sub-optimal decisions without the lower level detail. It also gives the rest of the org the feeling that the change is being done TO them rather than WITH them, which batters morale. Defining a Vision and Missions backed by data, then inviting business units to define objectives that help you on those Missions, devolves the detail design to the people with most relevant skill and knowledge and involves them, addressing all three of these issues. See also the VMOST model which describes this approach.

The most important question to ask (& answer) before undertaking any of the strategies outlined here is 'why?'. Why are these various businesses in the group? They were either acquired, started, spun out etc for a reason. What are these businesses purpose and strategic place within the overall group. If they were acquired / created to support other businesses or the group as a whole then their needs are subservient. If they exist as a vanguard into new markets, technologies etc then their needs are paramount. Each business will have its own reason for existing, and until you map that (& validate that it still applies), developing a group wide strategy is a fools errand.