The alerting framework doesn't log an error to the console when using the addLastRunError API

[mikecote]

I've been exploring our metrics in serverless now that we have the fix for the addLastRunError errors (#179551). I noticed we do not log the error message to the console, making it hard to perform analysis on our top reported errors.

Steps to reproduce

1. Apply the following code

diff --git a/x-pack/plugins/stack_alerts/server/rule_types/es_query/executor.ts b/x-pack/plugins/stack_alerts/server/rule_types/es_query/executor.ts
index 5e23bf9498e..548c36b9423 100644
--- a/x-pack/plugins/stack_alerts/server/rule_types/es_query/executor.ts
+++ b/x-pack/plugins/stack_alerts/server/rule_types/es_query/executor.ts
@@ -71,6 +71,8 @@ export async function executor(core: CoreSetup, options: ExecutorOptions<EsQuery
   let latestTimestamp: string | undefined = tryToParseAsDate(state.latestTimestamp);
   const { dateStart, dateEnd } = getTimeRange(`${params.timeWindowSize}${params.timeWindowUnit}`);

+  services.ruleResultService?.addLastRunError('some failure');
+
   const { parsedResults, link, index } = searchSourceRule
     ? await fetchSearchSourceQuery({
         ruleId,

2. Create an ES Query rule
3. Wait for it to run
4. Notice there are no error logs in the console while they are reflected in the UI
5. Change addLastRunError to throw new Error('fail');
6. Wait for the rule to run
7. Notice there is an error log in the console

Expected outcome

To log the error to the console similar to when an error gets thrown. A message could appear like this:

[2024-04-03T09:02:50.070-04:00][ERROR][plugins.alerting.es-query] Executing Rule default:.es-query:74ae32a2-f745-4137-9d9f-e1e6a4f9fcfe has resulted in the following error(s): fail

[elasticmachine]

Pinging @elastic/response-ops (Team:ResponseOps)