To add Blog page to Google Sites like this, you can follow these.
Google Sites doesn’t support blog feature, so we will use a free tool to do this.
#1. First, install this free blog widget
Click Create widget for free
#2. Choose your desired layout and click Continue with template
#3. Widget provides a lot of options, you can customize them. Then click Add to website for free .
Next, click Publish on top right.
#4. Choose Free Plan & Copy embed code
#5. Access Google Sites > Click Pages > Click Plus icon
#6. Choose Fullpage Embed
Name it “Blog” or any name what you want.
#7. Next, click Add Embed
#8. Click Embed Code > Paste the code you got in #4. Then click Next
It will show a preview window, click Insert
#9. Result