Just guessing, but the SABR work (i) pre-dated SVI, and (ii) spoke directly to FI quants with examples from there.
Also, based on many threads here, I am pretty certain the practical interest from traders in the model is almost purely, as you say, as an interpolation formula with some theoretical justification. Since there is a model underlying the interpolation, this adds another point in its favor vs. SVI which is very nice but more ad hoc.
Objectively, your point is well taken -- why bother with an interpolation formula that so easily breaks down (deep otm is not arbitrage-free) and is so much trouble (from the traders point of view) to correct. (Traders have little interest in the exact solns to the model, as far as I can tell, as they will tell you they don't take the model 'seriously'. What they really mean is "yes, the interpolation formulas have problems, but correcting them by solving the model exactly/numerically is way too much work!)
Finally, which is the cooler acronym?