Disclaimer: I do no have extensive SharePoint/.NET experience, mostly read through online Microsoft docs, so asking opinion in this forum.
I am modernizing 2 legacy microsoft apps. One is SharePoint 2013 site used for CMS and other is ASP.NET 2.0 web application having transactional frontend. New combined frontend must be JS based common for all screen sizes. Business wants most cost-effective solution, preferably on-prem. I am bit confused about the approach to unify both. 1) Upgrade path - Put ASP.NET into SharePoint farm and then modernize the UI pages. 2) Rewrite with SharePoint frontend - Build using SharePoint Framework (SPFx) which internally supports modern JS frameworks. 3) Rewrite without SharePoint frontend - Build a common JavaScript UI portal (ReactJS) and access the backend REST APIs (including SharePoint) sitting behind an API Gateway.