I would just like to point out with this one, we already offer a 'sane solution' to common pain points, and endeavour to improve that even more into the future. But the reality is, what you are doing at the moment isn't actually saving you hassles.
We offer many bug fixes between every version of the platform. If (for example) there are 500 lines of code between 2 given versions that have been changed, that's 500 times something may have been either a problem change, or something that is incompatible with your custom code (of which we have no control). If you don't upgrade 10 times, you are then looking at 5000 instead. So upgrading very infrequently, is actually very counterintuitive
On to the issues at hand. Your theme is indeed customised, as you can see here
This is the first place you should really be looking. You should be manually upgrading the templates that are modified between versions of the software, otherwise it will indeed cause you problems. You have modified code in the global template for example, which will affect every page on your site. We provide tools for checking the changed here
https://invisioncommunity.com/index.php?app=core&module=system&controller=plugins&do=diff
Just as a small example. Your last version was 4.7.2.1. If you compare that between then and now, there are changes on globalTemplate. So whatever those items change, will either break your site in some way, or not fix something that should have been fixed.
This is not to mention you currently have 12 x 3rd party applications installed and 11 x 3rd party plugins. These are all items in which you run at a risk of breaking your site. They should be fully tested between updates, and ensured to also be up to date as the authors are likely fixing things between updates. While not all will show if there is an update (you would need to check with the authors), there are at least 3 there which actively state they are out of date, so its very very likely they will be breaking things.
I completely understand that while running a business, you will have needs that come outside the box of the vendor in which you are using, and that you are using 3rd party and customised items to fulfil those needs. I also understand that you dont want the hassle of upgrading frequently, and so choose to upgrade very infrequently. But I'm a little confused as to how we can provide tools to assist you along with this process, as currently you appear not to be using the ones we provide.