There are a few relevant factors:
Use diagrams when they add value
I see plenty of formal writing that includes diagrams -- technical flow diagrams, trend graphs, timelines, resource-allocation charts, and more. The main question you should be asking yourself is: does this diagram add value? Does it make my point more clearly, compactly, or persuasively than words would? If so, consider adding the diagram. (I don't think your proposed diagrams add value, though. More on this later.)
But follow conventions
If you're writing an analysis piece on recent elections for the press (for example), charts showing voter breakdowns in the sectors or demographics you're talking about are probably normal, maybe even expected. If you're writing a journal article, conventions vary but you can easily find out what the norm is -- look at other published examples and see what kinds and quantities of diagrams are used. And if you're writing for your own blog, you can of course do whatever you want.
You mentioned cover letters and school applications. In my experience, diagrams are extremely rare in these contexts. (I don't see a lot of school applications these days, but I see a lot of cover letters and I've never, ever seen a diagram in one.)
Know when you should blaze a new path anyway
Based on what I've said thus far, you should be very, very reluctant to use diagrams in cover letters and school applications. But that doesn't mean never; there could be exceptions. If the diagram really is the best way to present relevant information, and doing so demonstrates a quality being sought (maybe you're applying for a design or marketing position), then you could consider adding diagrams anyway. Your application will stand out; you want it to stand out for a good reason and not a bad one.
Finally: about your diagrams
In the case of the examples you've shown here, I personally wouldn't include them. Your diagrams don't convey meaningful information; you seem to want to use them as decorations. The second one supplements what will presumably be text explaining how you work smart (so the diagram doesn't add anything); the first one is actually a bit of a mystery to me and I can't tell what point you're trying to make, but even if I could, you'd probably be better off making that point in words like other cover letters do.