Why do gram-negative bacteria attack the digestive system more than gram-positive ones?
Gastrointestinal infections can be caused by Gram positive and negative bacteria:

Gram-positive:


Staphylococcus aureus 
Clostridium difficile, botulinum and perfringens
Listeria monocytogenes
Bacillus cereus


Gram-negative:


Salmonella enteritidis and typhi
Campylobacter jejuni
Shigella
Escherichia coli
Vibrio cholerae and parahaemolyticus
Helicobacter pylori
Yersinia enterocolitica


Any part of the GIT can be infected by G+ or G- bacteria:


Stomach: S. aureus (G+), H. pylori (G-)
Small intestine: Bacillus cereus (G+), Salmonella enteritidis (G-)
Colon: Clostridium difficile (G+), Shigella (G-)


Certain bacteria typically affect only certain parts of GIT: H. pylori and S. aureus affect mainly the stomach; C. difficile affects mainly the colon.

Anyway, infections of the gastrointestinal tract are much more commonly caused by Gram negative than positive bacteria (Stat Pearls, 2019).

From the pathogenicity viewpoint, the main difference between Gram-positive and negative bacteria is that G+ bacteria produce exotoxins (which are proteins that bacteria secrete into their environment), while G- bacteria produce endotoxins (which are lipopolysaccharides as parts of bacterial walls). The differences between endo- and exotoxins (microbiologyinfo.com): 


Endotoxins are heat stable, so they are not destroyed by cooking, they don't need specific receptors, they don't trigger antibodies and there are no vaccines against them.
Exotoxins are heat unstable, they trigger antibody response and have specific receptors in the GIT.




More than 80% of skin infections is caused by Gram positive bacteria, such as Staphylococci and Streptococci (Clinical Infectious Diseases).

75-95% of uncomplicated urinary infections (Microbiology Spectrum) and most sexually transmitted infections (Clinical Laboratory Science) are caused by Gram-negative bacteria.

Respiratory infections are caused by both Gram-positive and negative bacteria (Lumenlearning).

So, the only pattern I see is that in dry/air-exposed areas (skin, respiratory tract), most infections are caused by Gram-positive bacteria, and in wet areas (gastrointestinal and genitourinary tract) by Gram-negative bacteria. This doesn't mean that there is any actual biological mechanism behind this pattern, though.