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One benefit of keeping the shipping out of the headline price is that you can advertise the products at a lower price on the likes of Amazon/Google, hopefully making users more likely to visit your store.
I guess it partly depends on the margins involved and also the average order quantity. If people are on average only buying one item, then they may be more likely to prefer free shipping. I don't know this for sure, but my gut feeling is that customers don't mind paying shipping for multiple items, but are less happy doing so for only one item.
Personally I have a spreadsheet that works out exact shipping using the weight of the product and then factors that in to the overall price, given the margin that I'd like to achieve. That's obviously much harder to do when the distributors don't give you the product weights/dimensions.