Yes !! Tomate à la croque au sel as we call'em !! First time I ate a tomato like that in the US, my american friends made so much fun of me "yo, no way brooo ! he's eating a tomato like an apple, da'fu !! Weird frenchman !!".
I still eat my tomatoes like that every so often, it's so good ! Although I think I end up eating quite a bit of salt when I do that which isn't so good...
In my head the biggest problem with this is that tomatoes are very liquidy and that seems like a very messy proposition.
Obviously, sliced tomatoes with salt is a relatively common breakfast side item, so the idea of just eating tomato with salt isn't that uncommon in the U.S.
Eat enough of them, and you'll develop the proper technique - do the initial bite slowly and carefully, while gouging the internal pressure. You might need to try another spot. Then immediately suck the interior. Then proceed eating them one pocket at a time, while occasionally sucking any free floating liquid.
Waffle House, one of the larger chains of grill made food offer sliced tomatoes as a side in place of hash browns. I have no found that offering elsewhere. So it makes me wonder how it made to the menu
Salt is fine... in general you're more likely to get too little salt than too much, and the difference in your blood pressure between the two isn't that significant.
I love tomatoes. They are seriously one of the best foods that exist, but I do find they can be messy, so I really enjoy a large plate of thinly sliced tomatoes with salt and pepper more evenly distributed as an alternative to this. Very cool to learn the French name for this though!
A few other things great with salt and pepper: grapefruit, watermelon, cantelope, cottage cheese, buttermilk.
If it's just large tomatoes, people often cut them into slices and then salt them. Or cube them along with cucumber, or halve them if they are cherry tomatoes
You add salt as you eat on the exposed portion. You can try putting salt on the initial bite spot, most of it will fall away, a bit will remain though.
I still eat my tomatoes like that every so often, it's so good ! Although I think I end up eating quite a bit of salt when I do that which isn't so good...