First off, AOZ has a wonderful page about titanium alloy uses in medical applications you should read and that most of my info is coming from:
I think you're having a couple of easy to correct misconceptions. First, titanium alloys are used for medical implants and replacements because they're lightweight, stronger by tensile strength than steel and judged to be completely inert and immune to corrosion by all body fluids and tissue, and is thus wholly biocompatible. It's also non-magnetic, meaning you can even go through an MRI machine with it! The range of alloys that we're able to produce is also key to the widespread use of titanium in medical applications as different alloys can serve their purpose as anything from hip joints to jaw reconstruction bases to spinal fusion devices based on their composition.
But cells don't exactly "stretch on it" and in fact in many operations additional material is used to ensure that the host body is able to function with the implant. This includes the use of polymethyl methacrylate bone cement to ensure that implant joints stay in place, and the use of roughened bioactive surfaces (hydroxyapatite) to stimulate the integration of the bone and surrounding tissue onto the implant.
As far as mechanotransduction goes, the problem is that it covers an extremely wide range of topics so ascribing it only one definition can quickly cause issues between studies. I usually go off of Nature's definitions for these kinds of things, which in this case says that "Mechanotransduction refers to the processes through which cells sense and respond to mechanical stimuli by converting them to biochemical signals that elicit specific cellular responses" (1). If you peek at the studies highlighted on google scholar for mechanotransduction you can see that the range of work is extreme and that's why ascribing the term one specific meaning at this point is unrealistic.