How to watch Apple TV+ shows on a TV without the Apple TV device?
You can watch Apple TV+ on the web at tv.apple.com.  From there, you can cast it to your Chromecast to watch on the LG tv.

LG has a few tv's that currently support Apple TV+ but there's only 4 of them right now and you need to use AirPlay 2 to send the content to the tv:


LG OLED (2019)
LG NanoCell SM8X series (2019)
LG NanoCell SM9X series (2019)
LG UHD UM7X series (2019)


As more manufacturers see if this is a success or not, you can expect to see changes in support.  

I would recommend signing-up for the 7-day free trial and use a web browser to cast it to your tv.  You can then decide if it's worth buying an inexpensive/used Apple TV 3rd gen to continue (or not if you don't mind using the Chromecast).

For more info on devices and shows you can visit Apple's overview page.