We are planning to relaunch our website and I have noticed that certain pages that receive good visibility are just too deep (4 and above). I'm planning to make these pages easier to access and reduce the crawl depth (while also deleting some and using rel=canonical accordingly) What I want to know is how the SEO performance of these pages will get affected if I decrease crawl depth.
Identifying pages that are more than 4 clicks deep and reducing that to a lower number is a good thing for SEO especially if those pages are valuable.
Adjusting canonical tags is not a way of accomplishing this though, canonical tags and click depth are not related.
Improved internal navigation through internal linking would be the solution. Adding category pages or linking from blog posts would be two examples of how you could achieve this.
You should also also be looking at entry points for authority on your site from backlinks. If these entry points are deep pages with a high crawl depth, then the authority will not spread across the rest of the site efficiently. Conversely if they are not the entry point, they are likely to receive much lower authority. These are broad statements, but without investigating the whole internal linking structure of your site there is only so much I can say.
Anyone saying that it doesn't, cannot fully understand crawl depth or SEO in great detail.
This blog covers how to setup what you need on most server types.
It does not affect SEO. SEO is about structuring and presenting information in a way search engines understand best. A page being deep in the site hierarchy has nothing to do with how well it is structured and consecutively, with its SEO score.