Handlebars is used for creating templates within your site, if that's part of your theme you may break functionality by removing it. Boomerang I'm not sure in your case but here are some tips (I'm the founder of the Shopify optimization agency Speed Boostr
1) PageSpeed Insights gives good general advice but you have to weed through irrelevant or inadvisable warnings if you're on an engaging ecommerce website that uses apps. Most engaging ecommerce sites I've seen score low with that tool (even billion dollar businesses). If you're interested in diving deep into that and how to improve the score with that tool this article will help: https://speedboostr.com/google-psi-for-shopify.
2) I would start with standard optimizations for Shopify sites. To do that, run your site through the Shopify Analyzer. That's a free tool my team built for the Shopify community. From there you'll get recommendations that you can take to your team to fix. If you need help, you can either contact us (speedboostr.com/contact) or go through our Shopify optimization guide.
3) To find code that is not needed but affecting performance, take a look in your major theme files (theme.liquid, product.liquid and its descendants, etc) and look for code snippets from old apps you've uninstalled. When you delete an app, Shopify revokes that app's access, so it's not able to clean up after itself if it added code to your site.
4) If you have code that is affecting your performance but is used in your site (like handlebars, or code from current apps), then it serves a purpose and shouldn't be removed. Analysis tools are just automated recommendations, they don't factor in revenue generating functions. My advice is to optimize what's possible but don't strip your site down just to get a higher score in any tool.