Using WordPress these days takes a lot of resources. Gone are the days of it being lightweight. If you start adding plugins that are dynamically adding content, doing tagging, pulling in images from external sources - fun stuff like that, then you need a decent server setup. Shared hosting on a budget just won't do it.
I don't think that HG are lying, but it'd be helpful if they were to separate out WP and other similar CMS sites from static HTML sites as the requirements are hugely different!
Users would be well served to follow HG's strong advice to use a good caching plugin but also to be aware that not all WP setups can work comfortably with a cache. Caching enables a site to work with much lower resource needs - at best, not far from a static site and, often, much faster page loads than WP normally can manage on a shared setup.
I use Dolphin and, in the past, Joomla. Users have the same complaints about these two platforms on shared hosting and for the same reasons.
Hostgator are great though if you are running a business dependent upon your sites because they offer an easy and inexpensive upgrade route from shared hosting through to dedicated servers. They handle the upgrade for clients such that the transfers are pretty much hands free and invisible to both the site owner and to visitors. That means one can start out with budget hosting and, as one's needs increase, move up to dedicated via VPS.
I used a dedicated server with them for years, after making a similar progression. I only changed over when I went to a provider who specialised in Dolphin setups, but there was no saving in money.
Sadly HG do not have much experience with Dolphin and this somewhat forced the issue. I regarded the cost, around $175 per month as a very low cost business expense and was very happy with the level of service until I found myself at the limit of their expertise with the arcana of Dolphin. By the way, there are people providing dedicated hosting at lower cost than HG but the shortcuts imposed by the lower costs are likely to provide a significant stumbling block for many people as the lower costs mean limitations to service level and quality, compromises on REAL server specs and the use of lower cost control panels.
Of course one should, ideally, monitor one's own resource usage but many do not. HG are pretty good at notifying account holders when there are resource issues; it is in their own best interests to do so as this is an ideal opportunity to sell an upgrade and, over they years, I have learned to follow the host's suggestions about upgrade needs!