Some interesting comments and suggestions - but there are problems & consequences if some of the suggestions were implemented.
So - just my opinions.
1. Lime Street site (at surface level) is almost impossible to enlarge to take full length HS2 trains. It will only be able to accept shorter length "classic compatible trains". An 11 coach Pendolino is about the longest train it will ever be able to accept.
2. Central High Level was too short to accept even something like a 9 coach Pendolino.
3. Exchange site is still largely useable (if you spend enough money), but any new line would almost immediately need a long tunnel to exit from the city central region. The frequency of local services on the Southport, Ormskirk or Kirkby lines is such that none of them will ever be suitable for fast, long-distance trains. 4-tracking as far as Ormskirk or Kirkby would also be very difficult & expensive.
So - if full length HS2 trains are ever to come to Liverpool, a completely new station - probably underground - is almost the only solution.
Station capacity. Yes, given enough money, Wapping Tunnel might be reinstated as part of a link from Edge Hill to Central Low Level. But, diverting locals away from Lime Street would lose connectivity with the long distance services to London, Birmingham, etc. - and I suggest that would be very unpopular.
And what would be the benefits for existing passengers - they would get less attractive trains with, probably, fewer seats than a Class 319 or 323 - and then be faced with the already overcrowded Central station. On a simple route network, a single island platform could cope with more trains per hour. But the "City Lines" network is anything but simple. Unlike much of the current Merseyrail Electrics lines, the City lines have complex interactions with other routes. Trying to push many of these extra City Lines trains into Central would wreck punctuality / timekeeping. Widening the tunnel at Central would be hideously expensive, and the entrance / booking hall area would also need to be enlarged. Continue to Moorfields, and you have long walks and 2 long escalators before you reach the exit - in a part of the city that does not seem very welcoming at night.
And what is the demand for linking services across Liverpool city centre ? The most popular destinations would probably be Chester & Birkenhead - but you would still have to change, at Central rather than Lime Street. Next - Southport, but that already crosses the city to/from South Parkway/Hunts Cross. And I doubt that there is much demand for City Lines passengers to travel to/from the Ormskirk /Kirkby lines (except once a year on Grand National Day)
So - a complex problem - with no easily affordable solutions - and a danger of making things worse, not better....
If nothing is done to relieve Lime Street high level, Liverpool is going to be completely stuck for evermore with its poor current selection of long-distance services.
The problems you set out do not require a new terminus, necessarily (and in any case the most obvious site, at Exchange, is about to disappear under a new office development).
What is needed is a combination of diversion of local services, with a high speed line (anathema is that apparently is to many on this forum) carrying the bulk of long-distance services. Broadly what Chester1 suggested, though I think you'd need a high speed tunnel to Edge Hill (assuming that's where the local trains would join the Wapping and / or Victoria tunnels to go underground).
If, for example, you confine most long-distance services to a bew tunnel leading east of the city (linking to a line to Warrington and all points east and south), and to the fast lines on the WCML to Edge Hill (for services via the Halton Curve), this just leaves trains towards Blackpool and Glasgow on local lines (perhaps they could eventually be placed on the by-pass line and routed north via a spur east of Huyton, or some link to the WCML further east). The local lines east of Lime St could then become full Merseyrail extensions which could be kept segregated from other operations except at their extremities (as happens now). A few residual regional services from Lime Street high level (eg semi-fasts to.Manchester via Chat Moss and the CLC) could easily fit around this, particularly if there was the odd bit of capacity enhancement on the local lines.
As Central Station needs to be rebuilt to handle its present passenger numbers anyway, enhanced usage in future can be taken into account in that, including some sort of fixed.link to Lime Street. The unpleasantness of the area round Moorfields is beyond the power of the railways to fix, but a comprehensive development of some of the mosr idiotically underused city centre land in Britain should.include a new and more user-friendly statoon.
A further issue which would need to be addressed would be freight. Unfortunately.some pillock(s) allowed the most obviously re-openable route to the docks, from Wigan via the North Mersey Branch, to be built over. Some.solution must however be possible to remove the need for trains to the docks to plough through the main approaches to the city for passenger trains.
Sorry to the Wirralians, btw, but given how much of a struggle it is to secure any improvement to rail services for the entire city region, there seems little prospect of restoring Birkenhead to the long distance network. Perhaps the better solution, until.such time as capacity needs are ao great that a revived Woodside becomes a necessity, is to restore 4 tracking from Rock Ferry to Hooton and introduce.a semi-fast to get people to main line termini at Lime Street and Chester faster.