Yes, building under the Park would certainly have saved effort and
money. When Oxford Circus station was being rebuilt for the Victoria
Line, the surrounding streets could not be closed to traffic, and
chief engineer HG Follenfant had to use raised platforms -- which he
called the Umbrella -- to lighten the traffic loads on the road
surface. It was that or have a lorry nose-dive into the new ticket
hall.

Another factor in the case of Green Park is that the station at
platform level is actually the old Dover Street. As the name suggests,
the original street level buildings were in Dover Street, with the
platforms not a huge distance away, horizontally. When the station was
re-built to incorporate escalators, the architects threw out not only
the lifts but the rest of the street-level station as well, moving the
exits to their current homes.

There is also the question of What's Down There. Subterranean London
is not unoccupied. It's a matter of legend that LU have only three
shots at goal when it comes to the exact positioning of a tunnel or
other excavation. If they're turned down, no reason is ever given.
They must simply guess again. If they're turned down three times, they
may not re-submit. So it's possible that this was a factor with the
new tunnels at Green Park. It's worth bearing in mind that the old
Down Street station isn't far away, and while there isn't any secret
infrastructure there now, there could be tunnels associated with it
that are deemed Not To Be Mashed Up.

Simon