I used something similar to that in mid-80s. I was a cheap young teenager who lived on a gravel road in the Ozarks and had a skinny tired 27" ten-speed which did get taken into the woods as well. My tires kept getting thorns and snakebites so I tried a hard foam "tube" from Walmart for $15. It looked a lot like that one in the link.
It solved the problem but was heavy and a lot harsher ride than on air tubes. There was a seam on the tire where it was glued or melted together to complete the circle. After a year or so, it split along that seam and a gap formed under the tire in one spot. Luckily the tube was hollow and it was the perfect size for a soft white bic pen case to fit in there. That caused bulging so I took the pen tube out, heated it and bent it in to an arc roughly the same as that of the tire. I reinserted the pen and it looked good so I rubber cemented it into place, let it dried and reinstalled on rim. It worked well enough until the bike got smashed somehow about 5 years later.
Many years later,I now ride tubeless tires on tubeless rims and Stan's ($$$ but hassle free). I find modern technology to be a much better solution but the other worked for my non-discerning hillbilly self at the time.
