Since the OpenText method isn't working for you, let's try something similar to this, which reads files using the built-in I/O methods (Open and Line Input). It should be quicker than FileSystemObject, and since you're working with raw text/data, you'll have more flexibility than just using Workbooks. OpenText.
We may need to add some conditional logic to detect when each "line" starts if your text file is jumbled (as it appears to be in the screenshots you've provided), but let's start by seeing how this works first.
Each line shall be written progressively from row 1 in Column A to rows 2 and beyond for each additional line.
Option Explicit
Sub f()
Dim ChatFileNm
Dim FF As Long
Dim destination As Range
Dim ctr As Long
Dim ln$
ChatFileNm = Application.GetOpenFilename(FileFilter:="Text Files (*.txt), >*.txt", Title:="Select Chat File To Be Opened")
If ChatFileNm = False Then Exit Sub
Set destination = Range("A1")
FF = FreeFile
Open ChatFileNm For Input As FF
Do Until EOF(FF)
Line Input #FF, ln
'Write the line in to the destination
destination.Offset(ctr).Value = ln
'Increment the counter
ctr = ctr + 1
Loop
'Release the lock on the file
Close FF
End Sub
Alternatively, build the entire text string from the file, and use the Split function with Chr(10) as your delimiter:
Option Explicit
Sub f()
Dim ChatFileNm
Dim FF As Long
Dim destination As Range
Dim ln$, txt$
ChatFileNm = Application.GetOpenFilename(FileFilter:="Text Files (*.txt), >*.txt", Title:="Select Chat File To Be Opened")
If ChatFileNm = False Then Exit Sub
Set destination = Range("A1")
FF = FreeFile
Open ChatFileNm For Input As FF
Do Until EOF(FF)
Line Input #FF, ln
'Write the line in to the destination
txt = txt & ln
Loop
'Release the lock on the file
Close FF
'Write to the sheet:
Dim lines
lines = Split(txt, Chr(10))
Range("A1").Resize(Ubound(lines)+1).Value = Application.Transpose(lines)
End Sub