Handle `RejectedExecutionException` everywhere #5384

Job	Run time
system-test (sentry-samples-spring-boot-opentelemetry, true, false)	5m 31s
system-test (sentry-samples-spring-boot-jakarta, false, true)	2m 46s
system-test (sentry-samples-jul, false, true)	1m 42s
system-test (sentry-samples-log4j2, false, true)	1m 32s
system-test (sentry-samples-spring-boot-opentelemetry-noagent, false, true)	5m 17s
system-test (sentry-samples-spring-boot-webflux-jakarta, false, true)	2m 41s
system-test (sentry-samples-spring-boot-opentelemetry, true, true)	5m 34s
system-test (sentry-samples-logback, false, true)	1m 27s
system-test (sentry-samples-spring-boot-4-webflux, false, true)	3m 22s
system-test (sentry-samples-console, false, true)	1m 34s
system-test (sentry-samples-spring-boot-4, false, true)	3m 0s
system-test (sentry-samples-spring-boot, false, true)	3m 11s
system-test (sentry-samples-spring, false, true)	2m 28s
system-test (sentry-samples-spring-boot-jakarta-opentelemetry, true, true)	5m 44s
system-test (sentry-samples-spring-boot-4-opentelemetry, true, true)	5m 39s
system-test (sentry-samples-spring-boot-4-opentelemetry, true, false)	5m 50s
system-test (sentry-samples-spring-boot-jakarta-opentelemetry-noagent, false, true)	5m 29s
system-test (sentry-samples-spring-jakarta, false, true)	2m 26s
system-test (sentry-samples-spring-boot-jakarta-opentelemetry, true, false)	5m 34s
system-test (sentry-samples-spring-boot-webflux, false, true)	2m 45s
	1h 13m 32s

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