fix(load/index): UserInfo dates as nullable yyyy-MM-dd HH:mm:ss strings

LastPlayTime and MissionChangeTime were typed as DateTime, which STJ
serialised as "0001-01-01T00:00:00.0000000Z" for a fresh viewer
(DateTime.MinValue). Prod's wire shape is "yyyy-MM-dd HH:mm:ss"
(no T, no Z, no fractional seconds) when present and null when
absent — verified against data_dumps/traffic_prod_tutorial.ndjson.

The .NET default format has a real chance of crashing the client's
DateTime.Parse path on any code that reads either field, and the
fields are presence-sensitive (NetworkTask-family Keys.Contains
followed by ToDateTime), so emitting the .NET default reaches the
client as a stale-but-present value.

Switching the properties to string? + FormatProdDateTime helper:
- non-default DateTime -> "yyyy-MM-dd HH:mm:ss"
- DateTime.MinValue -> null (omitted from wire via global
  WhenWritingNull policy in Program.cs)

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

SVSim.EmulatedEntrypoint/Models/Dtos/UserInfo.cs

@@ -1,3 +1,4 @@
+using System.Globalization;
 using MessagePack;
 using SVSim.Database.Models;
 using System.Text.Json.Serialization;
@@ -7,6 +8,9 @@ namespace SVSim.EmulatedEntrypoint.Models.Dtos;
 [MessagePackObject]
 public class UserInfo
 {
+    /// <summary>Wire format prod uses for the two datetime fields here. No 'T', no fractions, no zone.</summary>
+    private const string ProdDateTimeFormat = "yyyy-MM-dd HH:mm:ss";
+
     [JsonPropertyName("device_type")]
     [Key("device_type")]
     public int DeviceType { get; set; }
@@ -19,9 +23,15 @@ public class UserInfo
     [JsonPropertyName("max_friend")]
     [Key("max_friend")]
     public int MaxFriend { get; set; }
+    /// <summary>
+    /// Wire format <c>"yyyy-MM-dd HH:mm:ss"</c> (space-separated, no 'T', no Z, no fractions).
+    /// Null for fresh accounts that have never played — prod omits/nulls this rather than
+    /// emitting <c>DateTime.MinValue</c> with .NET's default ISO-8601-with-Z serialization,
+    /// which can crash the client's DateTime parser.
+    /// </summary>
     [JsonPropertyName("last_play_time")]
     [Key("last_play_time")]
-    public DateTime LastPlayTime { get; set; }
+    public string? LastPlayTime { get; set; }
     [JsonPropertyName("is_received_two_pick_mission")]
     [Key("is_received_two_pick_mission")]
     public int HasReceivedPickTwoMission { get; set; }
@@ -38,9 +48,10 @@ public class UserInfo
     [JsonPropertyName("selected_degree_id")]
     [Key("selected_degree_id")]
     public int SelectedDegreeId { get; set; }
+    /// <summary>Same format/null rules as <see cref="LastPlayTime"/>.</summary>
     [JsonPropertyName("mission_change_time")]
     [Key("mission_change_time")]
-    public DateTime MissionChangeTime { get; set; }
+    public string? MissionChangeTime { get; set; }
     [JsonPropertyName("mission_receive_type")]
     [Key("mission_receive_type")]
     public int MissionReceiveType { get; set; }
@@ -61,14 +72,17 @@ public class UserInfo
         this.Name = viewer.DisplayName;
         this.CountryCode = viewer.Info.CountryCode;
         this.MaxFriend = viewer.Info.MaxFriends;
-        this.LastPlayTime = viewer.LastLogin;
+        this.LastPlayTime = FormatProdDateTime(viewer.LastLogin);
         this.HasReceivedPickTwoMission = viewer.MissionData.HasReceivedPickTwoMission ? 1 : 0;
         this.Birthday = viewer.Info.BirthDate.ToString("yyyy-MM-dd");
         this.SelectedEmblemId = viewer.Info.SelectedEmblem.Id;
         this.SelectedDegreeId = viewer.Info.SelectedDegree.Id;
-        this.MissionChangeTime = viewer.MissionData.MissionChangeTime;
+        this.MissionChangeTime = FormatProdDateTime(viewer.MissionData.MissionChangeTime);
         this.MissionReceiveType = viewer.MissionData.MissionReceiveType;
         this.IsOfficial = viewer.Info.IsOfficial ? 1 : 0;
         this.IsOfficialMarkDisplayed = viewer.Info.IsOfficialMarkDisplayed ? 1 : 0;
     }
+
+    private static string? FormatProdDateTime(DateTime dt)
+        => dt == default ? null : dt.ToString(ProdDateTimeFormat, CultureInfo.InvariantCulture);
 }