the costs of fossilised software
When you said, "The site I hosted with that company didn't need an upgrade, it was running fine with almost no attention at all", what you really meant was: "I was not interested in bearing the cost of maintaining my fossilised software, and got a bit grumpy about the whole thing when my service provider made it clear that they were no longer interested in bearing the cost of maintaining my fossilised software either".
Centos 6 dates from 2011 and ships with packages and core o/s software from that era. It is completely true that an admin can stick with these operating system versions in the longer term, and this is a viable option which is regularly used by high value software installations. The problem is that there is a substantial cost in doing so and when you're maintaining low-cost sharing hosting services, costs matter quite a bit.
So it's not just about sticking with centos 6: it's about sticking with e.g. python 2.6, php 5.3, apache2.2 and a pile of other ancient software versions which are no longer supported by their developers and which are basically moribund. It's about committing to your user base that the hosting company will assume responsibility for fixing security problems in Django because the oldest maintained release of Django was released in 2015 and requires python 2.7. It's about not being able to provision any more customers on servers like this because you simply cannot sell python 2.5 and php 5.3 in 2017, which means that according as customers migrate off this platform, you're ending up with a much lower density of customers per physical server, which drives up the cost.
It's about the tool chains and automation software (e.g. puppet / ansible / salt) required to support cabinets full of older servers like that, which drives up the cost.
It's about having a support desk that can deal with issues relating to lots and lots of different software versions. It's about installing multiple software versions on these platforms and attempting to ensure that none of them are fighting with each other, which drives up the cost.
And it's about having to bear the clean-up cost when some customer who refuses to upgrade causes the shared hosting platform to become riddled with malware because this is inevitably happens, and drives up the cost too, and by a good deal too, because it needs to be handled by people with clue and people with clue cost money to hire.
So in case you're tempted to stick to your guns and moan that your fiver-a-month sharing hosting platform isn't offering you the long term stability that you want in order to continue to run your ancient and unsupported software, please bear in mind that there is a substantial cost to doing so, and that one way or another, someone needs to bear that cost, and that someone will be the customer because that's how supply of services works.
The only issue of relevance is deciding how that support cost is borne: whether through direct upfront cost increases, or by upgrading the platform, or by letting the platform rot and having customer attrition through neglect. Of these, the least bad long term option for the hosting companies is upgrading the platforms.
Otherwise I quite agree that it would be nice to have your cake and eat it.