The guts of 5e - specifically bounded accuracy along with combat, skills, saves, etc - is fairly well-suited for a low-magic campaign. Unfortunately the assumed setting in which the classes are based is not. Some class features could be preserved but the majority of the classes need to be redesigned for a low-magic setting. My vote would actually be throw them all out and start from scratch, preferably with an a la carte ability selection system for level advancement (so in a sense, classless).
There are a million and 1 ways to go about spellcasting too. It all depends on the style and feel you want. Is the ability to case innate? Does it still exist just rare? Did it ever exist in the world? Is it gone but the ability to learn available? Are there any additional costs (physical, psychological, social)? Does the Vancian system make sense for the world? One of my LONG-term projects is to create a toolbox for low-magic settings, sort of a re-imagined Grim Tales for 5E. Someday...
Another correlation I commonly see with regards to low-magic settings is making healing a lot harder and take much longer. While there's nothing wrong with that if that is your goal, low-magic doesn't have equate to ultra-realism (Conan for example). Realistic healing doesn't always equate to more fun at the table. It may result in your players become more cautious or not wanting to push the action per se unless they are fully healed. Downtime mechanics can mitigate this somewhat, but implementing systems like this should be done with kid gloves.
As far as Middle-earth for 5E goes, overall I was disappointed, at least as far as its contribution to a "generic" 5E low-magic campaign. The Fellowship and Journey rules are great, but are certainly not specific to a low-magic campaign. There is no magic system at all and the one class that is meant to serve as the "caster", the Loremaster, was a HUGE letdown for me. In general, I found the class design to be inconsistent.
So while it is possible to run a low-magic campaign with 5E, it requires a lot of resign, especially the classes. I will clarify that this is coming from someone who feels that simply capping the spellcasting level or giving out fewer magic items isn't the low-magic setting I'm interested in.