server: `TestAdminAPIEvents` takes too long to run in external-process mode

When `TestAdminAPIEvents` runs in external-process mode, the first test case takes around 40 seconds, while the rest run within the usual time. We need to investigate why this initial warm-up takes so long.

Example:

external-process mode

```
--- PASS: TestAdminAPIEvents (41.97s)
--- PASS: TestAdminAPIEvents/events?type=drop_database (35.26s)
--- PASS: TestAdminAPIEvents/events?type=create_database (0.00s)
--- PASS: TestAdminAPIEvents/events?type=drop_table (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table (0.00s)
--- PASS: TestAdminAPIEvents/events?type=set_cluster_setting (0.00s)
--- PASS: TestAdminAPIEvents/events?type=set_cluster_setting&limit=0&unredacted_events=true (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=0 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=-1 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=2 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=node_join (0.00s)
--- PASS: TestAdminAPIEvents/events?type=node_restart (0.00s)
--- PASS: TestAdminAPIEvents/events (0.00s)
```

vs

shared-process mode

```
--- PASS: TestAdminAPIEvents (6.57s)
--- PASS: TestAdminAPIEvents/events?type=drop_database (0.07s)
--- PASS: TestAdminAPIEvents/events?type=create_database (0.00s)
--- PASS: TestAdminAPIEvents/events?type=drop_table (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table (0.00s)
--- PASS: TestAdminAPIEvents/events?type=set_cluster_setting (0.00s)
--- PASS: TestAdminAPIEvents/events?type=set_cluster_setting&limit=0&unredacted_events=true (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=0 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=-1 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=create_table&limit=2 (0.00s)
--- PASS: TestAdminAPIEvents/events?type=node_join (0.00s)
--- PASS: TestAdminAPIEvents/events?type=node_restart (0.00s)
--- PASS: TestAdminAPIEvents/events (0.00s)
```

Related: #138912

Jira issue: CRDB-48302

Start by running the named TestAdminAPIEvents test in both external-process and shared-process modes, then inspect the test and its setup to compare the first subtest's startup path. Done means the external-process warm-up is explained and the test no longer spends roughly 40 seconds on its first case, with coverage for the regression if appropriate.

Written by the indexing model from the issue text.