fix(startup): multiprocess startuptime #26790

Problem:
Since 24488169564c39a506c235bf6a33b8e23a8cb528, the --startuptime report shows
two blocks of data. The TUI process and its embedded nvim process write to the
file concurrently, which may interleave the two startup sequences into the same
timeline.

Solution:
Report each process as a separate section in the same file.
1. Each process buffers the full report.
2. After startup is finished, the buffer is flushed (appended) to the file.

Fix #23036
Sample report:

    --- Startup times for process: Primary/TUI ---

    times in msec
     clock   self+sourced   self:  sourced script
     clock   elapsed:              other lines

    000.006  000.006: --- NVIM STARTING ---
    000.428  000.422: event init
    000.728  000.301: early init
    ...
    005.880  000.713: init highlight
    005.882  000.002: --- NVIM STARTED ---

    --- Startup times for process: Embedded ---

    times in msec
     clock   self+sourced   self:  sourced script
     clock   elapsed:              other lines

    000.006  000.006: --- NVIM STARTING ---
    000.409  000.403: event init
    000.557  000.148: early init
    000.633  000.077: locale set
    ...
    014.383  000.430: first screen update
    014.387  000.003: --- NVIM STARTED ---

1 files changed, 1 insertions, 0 deletions

diff --git a/test/functional/core/startup_spec.lua b/test/functional/core/startup_spec.lua
index f4a9c0c8d7..cc58226f48 100644
--- a/test/functional/core/startup_spec.lua
+++ b/test/functional/core/startup_spec.lua
@@ -73,6 +73,7 @@ describe('startup', function()
       os.remove(testfile)
     end)
     clear({ args = { '--startuptime', testfile } })
+    assert_log('Embedded', testfile, 100)
     assert_log('sourcing', testfile, 100)
     assert_log("require%('vim%._editor'%)", testfile, 100)
   end)