Hydrolysis of Sucrose over Sn1Ac or Sn2Ac
The mechanism supplied in the question statement is wrong according to this paper, where the evidence favored Carbon-Oxygen bond cleavage on the fructosyl-side instead of the glucosyl-side of the bridge Oxygen. As that reference states, "the initial products of the hydrolysis are a-D-glucose and a fructosyl carboxonium ion"



But don't feel bad, many textbooks have shown the same mechanism (detailed in the Discussion section of the paper). It's possible that the reaction mechanism changes with conditions, but I know of no evidence for that.

However, the same paper discusses that the mechanism was thought to depend on the type of sugar undergoing bond cleavage:



Results from a computational study show that SN1 mechanisms via a carbocation on fructose and glucose are both feasible, with the one on fructose slightly more likely.

Some other points:

1) The question statement refers to "primary" carbocations twice, and so do three of the comments. But in this reaction the 2 alternatives are a secondary or tertiary carboxoniun ion (for this terminology see Olah (1998) Onium Ions or ref 11 of the above paper).

2) It makes sense that an SN1-type mechanism would take place on the fructosyl-side of the bridge Oxygen since the resulting carboxoniun is tertiary rather than secondary.

Now, the real question is: How would you experimentally determine the mechanism? That is where the science comes in. The fact that so many textbook authors have presented this classic reaction with different mechanisms should be a big lesson to all students.  That expert authors would go to the extent of drawing detailed reaction mechanisms in a textbook that are not backed up by experimental reality, shows how mistrustful we should be of any non-experiment-based opinions. And this is for a classic, textbook example.  How much more unreliable "expert opinions" are for much more complicated things!