Finally spotted the plant whose spiky seeds keep puncturing bike tires. It's a low, creeping mat that can grow right next to the path and spread onto it, and the seeds grow in clusters. The dried seeds' spikes punch straight through, sometimes to the inner tube.
Unfortunately I have no idea what it is, since it doesn't match my previous guess (sand burr / sand spur) from what I can tell.
@abetterjulie Yes, that looks like it. Thanks!
And damn, puncture vine is an accurate name...
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