So will they? Okay lets roll up our sleeves and dig into this. S you site and every site out there that has more than 100 pages will have broken links. It will just happen. Now is that the end of the world? Not for one second. Google, Bing and all of the search engines out there know that this will happen and that the user will resolve to having a 404 page. Now should your 404 page rock and be amazing? Yes.
So if you have a link on your homepage linking to your top category page and that link is busted, well then there is an issue with UX and SEO. I look at broken links more to be a UX issue than an SEO issue with an internal link. Now say you have a rocking site, and that site has links pointing to it and you for some reason decide to turn off some pages and you just 410 them, well then you will have broken links and that is a bad SEO issue.
If you are “Killing” pages and you do not care about the SEO issues then I would 410 them. If you are wanting to preserve any and all link juice then you need to 301 the page somewhere that will make sense for the user to land on. If they were clicking on a page about “Men’s Shirts” please do not send them to the “Women’s Accessories” page. It just does not make sense at all.
So to close will a broken link harm your SEO? Only if you care about that page. If it is a product page that does not matter and you are churning pages at a high rate, then no. If it is a category page or hierarchy page, then yes. If you really care about everything 100% then instead of deleting a page, just send it to the hierarchy page above the page you are “Killing”