Tailscale must be properly configured on your client machine to access machines/ports on their respective private Tailscale network(s), setup of which typically requires administrative intervention. Without bridging to a public network, services exposed to the Tailscale network are not accessible publicly.
Tailscale does offer user-mode clients so it can be used similarly to SSH by those allowed to connect (I don't know how difficult user-mode Tailscale is without admin setup on various operating systems).
If your network firewall is preventing the tunneling process, then that's on you. and if its not on you and its a company decision then its VERY unlikely they'd be okay with cloudflare's publicly exposed ports.
Can you download and run Tailscale on a Windows client without Administrative access to install the software (setup the virtual NIC)? An SSH client is just a user-space app.
I appreciate that your approach is the more secure standard practice, yet want to make others aware of the edge cases here on a site called Hacker News rather than something like StackOverflow, where 'this is the way' reigns supreme.