Refactor connection handling #3762

Job	Run time
check (licenses)	37s
check (examples)	2m 2s
check (format)	14s
check (docs)	1m 21s
check (lint)	1m 42s
check (msrv)	2m 16s
check (spelling)	7m 0s
test (ubuntu-latest, nats-main)	11m 13s
test (ubuntu-latest, nats-dev)	11m 19s
test (macos-latest, nats-main)	11m 37s
test (macos-latest, nats-dev)	12m 43s
test (windows-latest, nats-main)	15m 52s
test (windows-latest, nats-dev)	15m 52s
	1h 33m 48s

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