Knowing what is inside the backup folder explains why the texts seem to vanish, and why an app has to read them for you.
The folder you see. iTunes from Apple's website saves backups under %APPDATA%\Apple Computer\MobileSync\Backup, while Apple Devices and iTunes from the Microsoft Store use %USERPROFILE%\Apple\MobileSync\Backup. Each backup is a folder with a long coded name. Inside are a few files with readable names, such as Info.plist and Manifest.plist, which describe the device and the backup, and a large number of subfolders with two-character names.
The files you cannot read. Every file the iPhone backed up is stored under a 40-character code instead of its real name, with no extension, in the subfolder that matches the first two characters of that code. A database called Manifest.db records which code belongs to which original path on the phone. Without it, a photo, a settings file and your message history all look alike: nameless blocks of data.
Where the texts are. The Messages app keeps its history in one database, sms.db, which the manifest lists under the phone's Library/SMS folder. It is a SQLite database built for the app, not a document. Messages, conversations, participants, reactions and attachments sit in separate tables linked by numbers, and dates are stored as counts of time since 1 January 2001. Newer versions of iOS often store the text of a message in a binary form rather than as plain text. Turning all of that into a readable thread means finding the file, joining the tables, converting the dates and decoding the text.
Encrypted backups add a layer. With Encrypt local backup on, the manifest and the files are encrypted with keys that are protected by the password you set. Nothing useful can be read without it, which is why BackupPaper asks for the password first and why a forgotten password leaves the backup closed.
Why iTunes will not show them. iTunes and Apple Devices make backups so they can restore a phone. They read the folder only to put it back onto a device and have no screen for browsing its contents.
What BackupPaper does with all of this. It reads the backup's description, uses your password to open the keys if the backup is encrypted, finds sms.db through the manifest and copies it into its own local cache, decrypting it there when needed. It reads the address book the same way to put names on senders. Those working copies are deleted when the backup closes, and the original folder is only ever opened read-only.
What it cannot change. If Messages in iCloud was on, Apple says the computer backup leaves out messages that already sync to iCloud, so sms.db may hold few or no conversations. BackupPaper shows what the database holds and does not bring back deleted messages.