If someone ate a venomous spider, would it still be fatal?

“There was an old lady who swallowed a spider,
That wriggled and jiggled and wiggled inside her.
She swallowed the spider to catch the fly.
But I dunno why she swallowed that fly -
Perhaps she’ll die.”

All 8 verses w. irritating music

I have very little medical/biological experience, but I would say “no” assuming that the spider never “bit” you inside your body. Most venom’s act once they enter the bloodstream. Swallowing a spider, while technically inside the body, is still outside the sterile confines of the circulatory system. I imagine that the acids and bases in the digestive tract would counteract the venom long before it ever got to the intestines and even if it didn’t the lower digestive system isn’t built to import foriegn chemicals into the blood stream. It probably depends upon the venom, and could probably make you quite ill, but I would imagine it would not be anything like actually being bitten.