Should it be tunnel? - No. A large proportion of the tunneling on phase 1 is purely for political reasons to appease NIMBYs. It's a real shame that we try to bury so much infrastructure for non-technical reasons.
Could it be tunnel? - Yeah
Cheaper - No. If it were cheaper it would be proposed.
Easier - Probably from a planning point of view but not an engineering view. If there was a continuous stratum of clay all the way then you could probably launch several Tunnel Boring Machines with a skeleton crew. 8 were used on Cross Rail for 42km. so you're looking at nearly 40 for London-Birmingham
Viable - Probably. Although the geology changes greatly from south to north. Different tunneling techniques would be required. Some of which are slower than snail's pace.
Cross passages every 500m would be required - Usually hand/small machine dug (That's 400 in all from London to Birmingham). Intervention/access/ventilation shafts every 1500m or so, or subterranean safe rooms would need to be dug.
I suspect there will be issues with aerodynamics in the tunnels of such length and the bores would need to be even larger to alleviate the "piston" effect.
HS2 guide for tunnelling costs (Excluding the railway systems fit-out)
https://www.gov.uk/government/uploa...file/434516/HS2_Guide_to_Tunnelling_Costs.pdf