My wife is travelling with our son to Sweden next month, going by train from St Albans to Gatwick Airport. Our son is two. He can walk, but he can also run off, so she'll be wanting him to be secured in his buggy when boarding and alighting - as she won't be able to get him on, away from the doors, then take on the folded buggy and a suitcase.
She is concerned that she won't have time to get the case on, or that he tries to get on/off the train when she's managing the case/buggy and falls.
I think for getting a baby on a train, being in a buggy is the safest way unless s/he is secured in another way, or carried.
However, once on we always take him out and fold the buggy, unless the train is virtually empty when it can be left by one of the doors (and moved if needed before approaching the next stop). I'm mindful of the fact that I don't want to ever be like some parents that do indeed push the buggy around and ask people to leave (or use the classic line as I once heard; 'Excuse me, you're in the buggy area' when pointing to the wheelchair symbol - and actually saying it to two elderly people who did actually move, but should have told the mum to ....
My wife is also worried about getting off at Gatwick, in that she'll take the buggy off backwards on to the platform, lock it, then need to get the suitcase off. She's fearful that the doors will close in this time*, even though I've said it should be there for about a minute at least - and is probably a staffed station so won't be dispatched until the doors are clear.
Thing is, I'm a lot more confident and wouldn't stress, but I know my wife might - and I can sympathise with other parents that will find something as simple as getting on or off a train a bit of a nightmare. And if you've had to wait for loads of trains to leave, I can also see the frustration turning into desperation. Given the mum mentioned here hadn't been trying to force the buggy through at the start, it does seem that this was the case and it's not just some arrogant mum that thinks she should be treated like royalty.
Believe me, I've seen those over the years. I even looked on Mumsnet once, and nearly had a heart attack! Thankfully, not all parents are like that.
* One reason is that she was with me when a FCC train driver closed the doors on people as they were still trying to get off a train at Hatfield (before the platform cameras were put in, mind) let alone others getting on. The doors did re-open, but I think this has made her worry about it happening again.