I think your opening post is a different question from what I was expecting when I opened the thread.
If you have to move locations - from one site to another, be that office, factory, building site, retail unit, training centre, hospital, people's homes, or whatever - during a day, so one working day consists of moving about, then I'd say yes, travel counts as work time.
But, if you are working in one place for the day, then no travel time is down to you. If you choose to apply for a job that involves you being a HQ twice a month, then that is your choice to apply for a job that is so far away from home.
I work at different venues quite regularly, and your first and last journeys of the day don't count as work time. Any additional journeys during the day, do.
This is something that has only come to more people's attention with the advent of more people wfh. So more people choose to either move away from where their work is, or people choose to apply for jobs that aren't near their home. All well and good, but it isn't the employer's responsibility to pay for your time or expenses when they aren't responsible for your choices.
I think that is different from you having a job in a workplace near where you live, and then being asked to visit a different site once or twice a month, as you originally applied for the local office. You (if I have read it right) are applying for a new job 3 hours away. That isn't the employer's decision.