DNS Cache or Router Issue Causing Inconsistent Results on Dynamic Websites?

Why are you focussed on DNS caching?  The DNS cache just keeps the IP Address associated with a particular hostname.  The browser's content cache is what speeds up page loading for static pages - not the DNS cache.  

But there are a lot of moving parts here, and your particular website is NOT well suited to measuring small network effects. 

The web site is cloud-hosted by Hostinger - it amounts to a virtual private server running on a collection of multiple connected physical servers sharing resources.  Their service uses server-side caching on top of the client browser caching.  It also has GeoIP support.  The first time I pinged it, I got an IP address located in the Ukraine.  Later on, it shifted to an IP address located in Arizona.  

ethanhales wrote:

Since websites that generate personalized outputs, such as birth-chart tools, zodiac wheel generators, ascendant calculators, and similar applications, rely heavily on real-time processing rather than cached content, even small networking delays become very noticeable to the end user.

Exactly backwards.  Since the web server has to generate the content dynamically in response to each request, the real-time processing load on the web server can drive the page load time.  When there is a spike in requests, there will be a spike in the response time.  

So there a lot of factors that will affect page delivery time here, and none of them are under your control.