Ranges are pretty common in APIs and both -Infinity and Infinity can naturally arise from one-sided ranges. Since they are absent in JSON, they are frequently replaced with null, ad-hoc sentinel values with uncoded assumptions (e.g. timestamps should be always positive) and missing fields.

I get that, but to go from "oh this won't be very common" to willingly "let's just leave this out" is something else. At least in my mind :) Or was it an oversight?

I suspect it was a bet on worse is better.

Whether it was a good bet is debatable, but given Crockford's focus on "try and leave out as much as possible" I can certainly see it making sense at the time.