What I don't get here is that the breaking of O-H bond (in Step 3) and not the C-O bond, even though the C-O bond is weaker than O-H bond.
As I can see in the 3rd step you’re referring to its a reversible process thus major and minor products are also formed so I think the breakage of the C-O bond would fall under one of the minor product, and the breakage O-H bond will fall under major product. I also think the oxygen's octet conditions would not be allowing it to do so.
I can see your concern about the Bond Dissociation Energy of the O-H bond being higher than that of the C-O bond.
A key point to consider here would be that the oxygen in question carries a positive formal charge, making it more electronegative and exhibiting a stronger -I effect.
This effect is experienced by both the adjacent carbon and the hydrogens; However, the carbon in question is a tertiary carbon and the three methyl groups stabilise the 'electron deficiency' of the carbon attached to oxygen by compensating with +I effects.
Hence, the hydrogen attached is electron deficient, and carries a partial positive; making it susceptible to attack by the lone pair on oxygen. Note that either Hydrogens could be attacked, and we get the desired product.
Hope this helps :)