How do I deal with the inevitable scenario when a table's bigserial runs out of ids but many of the old records are deleted?
If you consume 10000 sequence values per second, you'll have consumed 10000*3600*24*365 values per year, that is 3,1536E11. At that rate, it takes a couple of million years to exhaust a bignt sequence.

So you shouldn't lose any sleep over it.