The colder parts of the electrodes are not the ones where the arc root is, so how does the erosion there affect the discharge ?
The electrode lead in (support) wire is etched away and so breaks off. The same mechanism as in heavily underdriven halogen incandescent lamps...
And needed to note, in the CMH's the halide salts slowly dissolve the arctube wall material, till a leak path is formed. Another mechanism, where the halides shorten the life.
In real decent products the designers are well aware about all these mechanisms involved and so they are virtually perfectly balanced to yield maximum performance (longest life time vs maximum efficacy and color quality,...), of course the trade offs are made according the need for the given application (application requiring high color quality is tuned in a different way than another with the highest efficacy, or another where the lifetime is the dominant parameter).
The good advantage of these mechanisms is, they are (relative to others) perfectly predictable, so do not interfere that much with the reliability (so very minimum failures with reasonable maintenance group relamping schedule, with minimum wasted remaining life of the individual bulbs; do not confuse limited lifetime and reliability, these are way different things in the terms of installation management)