The short answer
WP Engine and Kinsta are good managed WordPress hosts. What you buy from them is the hosting platform: servers, caching, a CDN, backups, staging and round-the-clock support for that platform. Looking after the site that runs on it is still your job, or your developer's. That means plugin and theme updates, checking the site after an update, fixing what breaks and making small changes.
WhatArmy does both. We host the site and maintain it, or we maintain it on the host you already have. Care plans are $150, $300 or $500 a month, month to month. Hosting with maintenance included starts at $125 a month, billed annually.
If you already pay a developer you trust to look after the site, a host on its own is a sound setup. If nobody owns the site after launch, that gap is what a care plan covers.