Nicely put, though interestingly the hydration doesn’t even need to occur through serpentinisation or other hydrous minerals. In fact, serpentine and various other ‘dense hydrated magnesium silicate’ (DHMS) minerals are unstable above 1000° C and break down into other things; this translates to a depth of about 200 km in the mantle, which is only about half of the upper mantle. DHMS breakdown is a large part of what releases water from the downgoing slab in a subduction zone, which then fluxes into the overlying mantle wedge and allows melting to occur, thus creating the volcanic front associated with subduction zones.
General idea shown here for those not familiar.As for the rest of the mantle, the lower mantle has a lot more depth (you don’t get to the outer core until ~2900 km) and it’s believed to also convect. It manages to do so because olivine, pyroxene and their equivalent high-pressure phases (eg. β-olivine or γ-spinel aka ringwoodite) can accommodate H₂O into their crystal lattices even though they are anhydrous minerals (whereas serpentine is not serpentine without H₂O as part of its structure). How it works for olivine, pyroxene etc: Rather than having dedicated OH^- sites like hydrous minerals do, direct replacement of Mg²^+ with 2H^+ occurs in Mg-silicates; or coupled replacement of H^+ and Al³^+ replaces Si⁴^+ in Al-bearing silicates. Olivine may also accommodate OH^- in place of O atoms.
The accommodations are not exactly huge: β-olivine has been shown to incorporate up to 4000 ppm H₂O (0.4 wt%), but over the whole mantle this adds up to create the viscosity change that you described, making the mantle malleable enough to undergo constant continuous deformation with applied stress, ie. it convects despite being solid.
Perhaps also worth mentioning how mantle melting that occurs below
mid-ocean ridges essentially transports water from the mantle into the melt generated and much of this volatile content is then degassed when the most buoyant portion of melt is erupted onto the seafloor. The residue of (mostly) olivine and pyroxene that never got melted in this generation of new oceanic lithosphere (ie. the stuff sitting underneath the newly formed basaltic crust) is now effectively dehydrated and so is more rigid than before, leading to the mechanical boundary between lithosphere and underlying asthenosphere. So the whole dehydration process serves to reinforce plate structure as the plates are being made.