The "risk adverse" nature, I can't belive I just said that about a bank, will become a problem.
You might be surprised by the number of subtle bugs lurking in corners left out as "implementation detail" by standards.
I've learnt to treat browsers as capricious genies that stick to the letter of the standards, but inevitably screw you over by inconsistently doing the opposite of what a sane person would have regarded as implicit.
There should be very little risk, but the fact is, there is actually a great deal of risk. With my professional web developer hat on, the "evergreen" browsers are the best advertisement in the history of computing for why organisations that need reliability in their IT systems should be wary of automated updates controlled by outside parties.
Browser updates break stuff all the time. And not just obscure things, though there are plenty of those. On the small scale, I've seen Chrome updates break basic page layout, and rendering styles as simple as rounded corners or shadow effects. On the larger scale, Chrome sometimes removes entire chunks of functionality, like support for important plugins. New features are often the worst, and it's particularly insulting to be told we should all use HTML5 feature X or JS feature Y instead of plugins, when the reality is that the new version still isn't up to doing what the plugin did a decade ago.
Businesses don't care that it's more convenient for the browser developers if everyone rearranges their entire work schedules to keep up with the latest "living standards". Businesses just want software that works, and having found it, they will go to extraordinary lengths to continue using it rather than playing the upgrade lottery. And given the track records of most of the upgraders in this game, frankly, it's hard to blame them.