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.
the tailscale devices you see are only accessible by other devices on the same tailscale network.
S/he's talking about accessing those machines from OUTSIDE that network. That's what would require admin intervention. So for example if I have a webserver on my home LAN that has Tailscale installed and authenticated, then sure, I can access that webserver from any of my other Tailscale devices from anywhere. But if I want a friend to be able to access that webserver without first being authenticated to the Tailscale network... Do you see the problem, yet?
for starters, what you're describing is a load balancer. those already exist and are trivial to setup.