Liverpool doesn't need HS3 to connect to Manchester

Liverpool does not need HS3 to connect to Manchester, it needs it to connect to Leeds & Hull. It is not that getting Liverpool to Manchester down to 20 minutes wouldn't be useful, but the linkage between the city centres is not the only measure of connectivity.
Both Liverpool and Manchester City regions have large hinterlands with several centres of economic activity. There is no point measuring simply the connectivity between the two Centres, the interconnectivity must take into account the linkage between the sub nodes. While the linkage between Ellesmere Port and Bolton may not be as important as the Liverpool to Manchester link, it is not worthless and is one of many sub-node interconnections.
One of the longer distance routes available from Liverpool Exchange was the link to Bury via Liverpool and Bury Railway, unfortunately, this was cut back to Bolton in 1970 though most of the alignment is still in place and a 4.5-mile section would allow a link to Manchester Metrolink for line into Manchester City Centre.

Liverpool & Manchester rail linkages

The building of the Ordsall Chord seems to be funding and going ahead if coupled with that Merseytravel's Airport spur was constructed and the Warrington Altrincham line reinstated. Then an outer loop could be formed, providing direct links from,  Say Widnes to Bolton/ Altrincham and Bootle as well as providing links to between Liverpool, both Manchester's stations and Stockport.
A link from the Northern Line from Bootle via the stock interchange to St James and the Wirral would push the loop out to Ellesmere Port and Runcorn. Though a reverse at Liverpool could do the same.

Rail Details of Liverpool

A map of Liverpool with some of the changes proposed in Merseytravel's 30-year plan.

Rail Details of Manchester

Details of the Ordsall chord.

Warrington rail details

The red line departing right is the disused Warrington and Altrincham Junction Railway
all the other lines are intact and in use though the green line entering left is only used for freight and connects to Ditton Junction.

Timperley rail detail

This is Skelton Junction with the line entering from the upper left is the closed Glazebrook East Junction to Skelton Junction Line while the red line entering mid-left is the closed Warrington and Altrincham Junction Railway.

Tram Train

Some parts of the network are now operated by MetroLink but use old rail alignments. Converting these to Tram-Train allow connections to places like Prestwich direct from Liverpool.