Would I lose anything career-wise if I dropped a sysadmin course for a web development course?

It depends on what period you will do it.If you are at the start of your career, I think its ok.If you are 10 years sysadmin, this mean you must restart in onther area, the area of Web.I forced from Oracle Certified DBA to become CMS-techy, because I migrated to another counrty, due to economic crisis in Greece (2014)