iOS responsiveness woes

I am running U-Design theme on 4.4.2 and attempting to launch a new version of a site. The new version has existed for some time, and I’ve had to retool it somewhat to make it mobile-responsive.

Everything looks good across all the Android browsers I’ve tested. However, Safari and Chrome for iOS seem to respond differently to it by the hour. Yesterday, I installed the Average Mobile Detect plugin, which seemed to fix that. But, within a few hours, iPhones 5 and 6 were encountering both new glitches and the same old glitches I thought had been fixed, annoyances such as double menus and elements scattered all over the place.

Here is the site:
https://joeyskaggs.com/devsite

Have you had luck with particular solutions for iOS responsiveness I may not have yet tried?