How can I copy entire chain of text messages (not just a single one) with the stock Android messages app?

Stock SMS app on Pixel 4a does not have this option and you need alternative methods.

Use SMS Backup and restore

• Option 1 : Take a complete backup of your SMSes and from the app see "Backed up (time)" and tap view details. It will list SMS Backup. Tap on that to view conversations and all your conversations are listed. From the 3 dots menu against the conversation you want select print conversation and it will be printed as a PDF file with date and time stamps. Example below (You can get sent and received. I chose a conversation that doesn’t reveal personal details). From the pdf you can copy and paste wherever needed.

• Option 2: From settings → backup settings → selected conversations only, choose the conversation you want and repeat steps in Option 1 to get a print, share or copy to clipboard.

I am not affiliated to this app but it's likely the best of it's kind.