I suspect the observed outcome is a combination of two major factors.
First, it is indeed expected that siloxy substituents are poorer electron donors via mesomerism, compared to alkoxy substituents. One indication comes from the comparison of the pKas of the parent alcohols and silanols. For example, trimethylsilanol has an aqueous acidity constant approximately eight orders of magnitude larger than the carbon analogue, tert-butanol (pKa 11 vs. 19, respectively). This suggests the conjugate base is being stabilized by the silicon, and pulling the electron density away from the oxygen atom. In the past this was attributed to partial $\pi$ bonding between the 3d orbitals of silicon and 2p orbitals of oxygen (a type of hypervalency), but this is now not thought to be an accurate explanation. Instead, it comes down to hyperconjugative effects, where the oxygen lone pairs donate into the Si-C $\sigma^*$ orbitals.
Secondly, one could argue that the tert-butyldiphenylsiloxy substituent is less sterically demanding than may be expected from intuition. Organic chemists spend a lot of time dealing with p-block elements from the second row of the periodic table, which are anomalously small due to the lack of radial nodes in the valence 2p orbitals. Most atoms are rather larger. Typical Si-O and Si-C bond lengths are ~40-50% longer than the C-O and C-C equivalents. This means that the apparently enormous amount of steric bulk from the simultaneous tert-butyl and diphenyl substituents on silicon is actually spread out of a considerably larger volume, weakening their steric hindrance effects. A very crude attempt at visualizing this difference is shown in the structures below. The left structure has default bond lengths between all atoms, whereas the structure on the right has had the Si-X bond lengths increased by 40%, to more accurately reflect the increased size of the silicon atom. Note the increased space surrounding the position opposite to the methoxy substituent, even in the most crowded conformation. Of course, this is a 2D representation of a 3D structure, so in reality there is an extra spatial dimension further diluting the steric bulk. Roughly speaking, the silicon atom causes the bulky substituents to be diluted in approximately $1.4^3=2.74$ times as much volume.
