fix: race on storing schema engine last changed time

[harshit-gangal]

Description

This PR addresses a race condition in the Schema Engine related to updating the last changed time. The issue arises because this time is updated by both:

1. A ticker that periodically reloads the schema and updates the timestamp.
2. The VStream tracker process, which also modifies this value.

Logs from CI:

==================
WARNING: DATA RACE
Write at 0x00c0003d87c0 by goroutine 1861:
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Engine).reload()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/engine.go:619 +0x28f9
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Engine).ReloadAtEx()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/engine.go:393 +0x2a4
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Engine).ReloadAt()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/engine.go:373 +0x1e3d
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*vstreamer).parseEvent()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/vstreamer.go:550 +0x1dda
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*vstreamer).parseEvents()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/vstreamer.go:381 +0xd55
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*vstreamer).replicate()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/vstreamer.go:[211](https://github.com/vitessio/vitess/actions/runs/13694130321/job/38292683920?pr=17907#step:10:212) +0x27d
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*vstreamer).Stream()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/vstreamer.go:190 +0x2e9
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*uvstreamer).Stream()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/uvstreamer.go:438 +0x5e9
  vitess.io/vitess/go/vt/vttablet/tabletserver/vstreamer.(*Engine).Stream()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/vstreamer/engine.go:280 +0x27c
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Tracker).process()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/tracker.go:132 +0x646
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Tracker).Open.gowrap2()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/tracker.go:86 +0x4f

Previous write at 0x00c0003d87c0 by goroutine 173:
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Engine).reload()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/engine.go:619 +0x28f9
  vitess.io/vitess/go/vt/vttablet/tabletserver/schema.(*Engine).Open.func2()
      /home/runner/work/vitess/vitess/go/vt/vttablet/tabletserver/schema/engine.go:275 +0x4f
  vitess.io/vitess/go/timer.(*Timer).run()
      /home/runner/work/vitess/vitess/go/timer/timer.go:111 +0x167
  vitess.io/vitess/go/timer.(*Timer).Start.gowrap2()
      /home/runner/work/vitess/vitess/go/timer/timer.go:85 +0x44

Backport Reason: CI failure, leads to wait time for PR to be merged.

Related Issue(s)

Checklist

• "Backport to:" labels have been added if this change should be back-ported to release branches
• If this change is to be back-ported to previous releases, a justification is included in the PR description
• Tests were added or are not required
• Did the new or modified tests pass consistently locally and on CI?
• Documentation was added or is not required

Deployment Notes

[vitess-bot]

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 67.56%. Comparing base (c710c81) to head (3c6d55d).
Report is 3 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #17914      +/-   ##
==========================================
+ Coverage   67.55%   67.56%   +0.01%     
==========================================
  Files        1597     1597              
  Lines      259769   259769              
==========================================
+ Hits       175489   175521      +32     
+ Misses      84280    84248      -32     

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