The public's scepticism is quite right because outside of London, around the rest of the country, there are many rail and infrastructure improvements which cost a small fraction of that, just noise in the scheme of the HS-2 budget, which are popular with local communities but go absolutely no-where for decades because the funding isn't available.
So we have a general public who had it drummed into them for so long that money simply isn't available for even minor things in their communities, who then suddenly find there's a near bottomless pit of money for HS-2. Their local council can't afford to fix the bus shelters, but the bus services have been cut to the bone anyway, the roads are full of pot holes and there isn't even a normal train service to their local towns, but suddenly central government has shaken the magic money tree and hey there's £100Bn for a new high speed rail line. There's nothing for you and your community, but heck it's only £150 a year per person.
The whole line about 'releasing capacity' is fundamentally flawed, firstly because there's no direct capacity effects for large parts of the country outside the HS-2 catchment area, secondly because HS-2 by-passes many towns along its route and that extra capacity is coming from removal of services which currently stop at some of these towns (some towns will get a worse service post HS-2) and thirdly because there's the naivete of the HS-2 enthusiasts in believing that the £100Bn spent on HS-2 won't be part funded from funding cuts to the rest of the rail network. Ultimately there's only so much rail funding to go around and a sizeable chunk of that £100Bn for HS-2 will come from the existing network, so there won't be the cash to fund some huge growth in new services from al of the 'released capacity'.
HS-2 is an example of the huge disparity we see, where local projects get starved of investment but major government project get blank cheques.
But it's not a "blank cheque" for HS2. It is financed by borrowing set against its future revenues, underwritten by the Tresury. Not that most of the public understand that, of course.