It probably is possible, but it is definitely not likely or probable. Snakebite deaths run to about 12 per year, out of thousands of bites (I am sorry I do not have more precise figures - perhaps someone else does).

Cut and suck went away because often the therapy was more damaging than the original bite. Evidently there are snake bites and then there are snake bites. Some bites miss completely, while some are "dry bites," in which the snake does not inject any venom. In this case, having been run over by a baby carriage, Mr Snake was apparently rather POed and delivered a good load of venom.

A big variable is the size and general health of the victim. Small folks, the elderly, and those with compromised health are more risk. Another factor is the species of snake. Some are far more toxic that others, per volume of venom, by a factor of 100 or so.

It is much easier to act and do something, anything at all, if the victim is under stress, but, yes, keep the victim calm and get anti-venom ASAP. Note that the ambulance was there within twelve minutes. In the wilderness, obviously, it would take a bit longer.