Several popular quality plugins require a PHP version higher than 5.2. When using an outdated version, you do not have access to many of these solid plugins. Therefore the experience will be much more seamless on a supported version.

Might be interesting to look at the top X plugins and see how many of them require a more recent version.

Here is a list of the top 120 plugins: http://wp-info.org/plugin-top120/

Requires 5.3+
https://wordpress.org/plugins/w3-total-cache/
https://wordpress.org/plugins/backwpup/
https://wptavern.com/buddypress-2-8-will-bump-minimum-php-requirement-to-5-3
https://mainwp.com/system-requirements/
https://ithemes.com/codex/page/BackupBuddy_Specifications:_Server_Configuration_and_Minimum_Requirements

Not in Top 120, but also requiring 5.3+:
https://wpml.org/home/minimum-requirements/
https://podlove.org/2014/08/14/podlove-publisher-2-phasing-out-php-5-3/
http://wp-tiles.com/docs/upgrading-from-php-5-2-to-php-5-3/
https://wp-types.com/toolset-requirements/
https://cometcache.com/blog/new-minimum-php-version-php-5-4/
http://wp-cli.org/#installing
https://interconnectit.com/products/wordpress-performance-profiler/
https://www.getmotopress.com/plugins/
https://eventespresso.com/requirements/
https://thimpress.com/product/wp-pipes/
http://pdf-light-viewer.wp.teamlead.pw/wp-content/plugins/pdf-light-viewer/documentation/index.html
https://www.oasisworkflow.com

Requires 5.6+
WooCommerce Stripe
https://cinchws.com/woocommerce-stripe-plugin-now-requires-php-5-6/

Devs are using Composer sometimes, which itself is requiring 5.3.2+
https://getcomposer.org/doc/00-intro.md#system-requirements

(This list is obviously not complete. I've not checked all of the Top 120 plugins. Just a sample of some plugins requiring 5.3+)

I don't think the wp-info.org top 120 list is accurate. For example, it's missing WP Forms, which has 500k+ active installs. https://wordpress.org/plugins/wpforms-lite/

Also worth noting that some plugins recommend higher versions, even if they don't require them. WooCommerce recommends PHP 5.6+. And of course, Yoast SEO is actively pushing users to update. These plugins are both in the top 10.

@JDGrimes Yes, that's true. BackWPup is working with PHP 5.2, but then it does not support all backup methods.

@mindctrl Nice catch. I ping the author of the page.