Reducing your website to something that can essentially be served straight out of s3 buckets + lambda functions for the few interactive elements your site has (e.g. newsletter subscription, contact form) will reduce your hosting bill and attack surface, but increase operational complexity.
The key thing behind the "headless CMS" fashion is that most people severely overestimate the amount of cost reduction and underestimate the impact of something breaking down in the backend side as well as the engineering cost.
As if "devops" were a job... at most companies the sad reality is that the ops team gets fired and the developers who often enough don't even have more understanding of a Linux shell than ls/cd/cat/rm get told "you're also doing ops now, have fun!".
Inevitably, issues will plop up - hacks, data loss, site going down over the weekend because there's no 24/7 on-call any more - and then management comes down to the devs and whines "what's the cause of problem X"... and fires the rest when they say "we're developers, not SREs and server administrators".
The key thing behind the "headless CMS" fashion is that most people severely overestimate the amount of cost reduction and underestimate the impact of something breaking down in the backend side as well as the engineering cost.