You asked how we remotely access servers, and I work as a system engineer and server admin for a company that runs servers all across the globe, so remote access is very much a requirement. Here are my methods.
IMM/IMPI for remote power management and BIOS access. ESXi console for VM management (usually with Ansible, using SSH). SSH for VM access (Ansible again). A few servers have VNC access (mostly the Windows servers *shiver*). And finally, some of our systems run their services with a web page for admin.