Fix game log run lifecycle (#605) #1268

Job	Run time
Game capture smoke / Windows	1m 46s
Release smoke / windows-latest	1m 20s
Python 3.11 / ubuntu-latest	2m 0s
Godot tests / Windows	2m 12s
Stale server smoke / Linux	35s
Game capture smoke / macOS	1m 54s
Release smoke / ubuntu-latest	48s
Godot tests / macOS	2m 9s
Release smoke / macos-latest	56s
Stale server smoke / Windows	1m 9s
Stale server smoke / macOS	50s
Game capture smoke / Linux	1m 2s
Godot tests / Linux	1m 14s
Godot tests / Linux (Godot 4.3)	1m 9s
Python 3.13 / ubuntu-latest	4m 16s
	23m 20s

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