What you give up when you leave the theme
The theme editor goes away. That sounds minor until your marketing lead wants to move a banner before a promotion and finds the queue now runs through a developer. For plenty of teams that single change costs more in a year than the hosting they saved.
Your apps need auditing one by one. A large share of the Shopify app market installs by dropping blocks into your theme, and with no theme there is nothing to drop into. Some of those apps have a headless path, some have a workaround, and some you will be rebuilding, which is a line item people forget when they compare a headless quote against a straightforward Shopify build.
Multi-language and multi-country stop being a setting. Shopify's market configuration still exists, but the routing, the locale detection and the country selector become yours to build, and Shopify's own guidance warns that page caching and search engine crawlers make the naive approaches unreliable.
The good news deserves stating: checkout stays Shopify's. You are not rebuilding payments, and Shop Pay keeps working.