wellington between bank and elgin is not used for vehicular property access with the exception of a single federal govt lot (adjacent to the old US embassy). It is bordered exclusively by federal properties. All hill vehicle access is blocked off by security except for ceremonial uses. Actual hill access is via the bank street entrance. Almost all south-side federal buildings are served by north-south crosstreets.
The primary use of wellington in this area is for through-traffic and tour busses. Through traffic can easily be rerouted to queen and albert/slater via bank and elgin. The capacity will be there once the confederation line is open.
Also, there's probably some sort of security benefit.
Just do it.
(here are some rudimentary drawings of what it would look like).