Ok, so, some updates in case anyone has this problem:
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Upgrading to 5.1.2.2 had no effect whatsoever.
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The wiki article linked by Lupo (OpenGL - The Document Foundation Wiki) did actually help me out, though – I put the registry fragment is said to download into a .reg file, double-clicked it, and all my problems went away.
That article also made me think of something. The system I am running LO on is a virtual machine (Workstation 11). When I created the virtual machine, I clicked the “accelerate 3d graphics” toggle in the graphics card setup which is not something I normally do. Windows, and every other application I use, seem fine but I suspect that LO is the only OpenGL application on my system. Perhaps the virtualization driver does not support OpenGL well (or at all).
Even if that is the case, though, I would expect LO to do something sane – like check if there is support available before assuming there will be.
I did install the OpenGL Extensions tool and the text the WIKI asks for is:
Renderer: Gallium 0.4 on SVGA3D; build: RELEASE;
Vendor: VMware, Inc.
Memory: 1024 MB
Version: 2.1 Mesa 10.0 (git-5da4fa2)
Shading language version: 1.20