Would it? How much would it have saved?
https://assets.publishing.service.g...file/434516/HS2_Guide_to_Tunnelling_Costs.pdf suggests that in 2015, a 7km twin bore tunnel was coming in at around £500 million. So probably a bit more today as everything has gone up.
Though in terms of network effectiveness, if as a country we have say £100 billion to spend on high speed rail. That money might have been better spent on a 7km tunnel getting the line into the centre of say Sheffield from the South, than in the middle of nowhere. Elevated sections might be a good way of cost saving in urban areas, such as for a line from Sheffield-Meadowhall, East Manchester (where proposed) and the Southern/NPR access to Leeds. But were they considered for Old Oak Common to the M25, I'm assuming they're not practical for Euston to Old Oak Common. In the case of Liverpool, I actually think Lime Street is ok for size, particularly if extended onto adjacent railway land, the biggest difficulty is the congestion through the Edge Hill Tunnels. Maybe the station upgrades will include and extra bore or 2 though Edge Hill, though there must be a point at which a separate section becomes easier.
Thinking of the Services Liverpool would have Post HS2: This seems roughly the service pattern to expect of have I missed anythin
HS2 2 per hour to London via Warrington BQLL
NPR 4 per hour to Leeds via Warrington BQLL
WCML 1/2 per hour to London via Runcorn (more stops south of Crewe, e.g. Rugby, Milton Keynes)
7/8 tph, dedicated 4 platforms by taxi rank and southern bores of Edge Hill Tunnel direct to Wavertree flyovers.
LNW 2 per hour to Birmingham via Runcorn calling at South Parkway
TPE 1 per hour to Edinburgh/Glasgow via Preston (stopping at Wigan only)
Norther Connect 1/2 per hour to Blackpool via Wigan (stopping at Huyton and St Helens)
EMT/TPE 2 per hour to Sheffield via Manchester (could be semi fast via Newton le Willows)
6/7 tph
CLC 2 per hour, stopper to Manchester (Oxford Road or Piccadilly) via Liverpool South Parkway and Warrington Central
Chat Moss 2 per hour, stopper to Manchester Victoria via Newton le Willows
Wigan 2 per hour stopper via St Helens
6 tph
I think this is probably doable, 12/13 tph down a double track tunnel for the Slows. Wavertree needs some work with flyovers and maybe stealing a bit of the depot and some optimising further along.