I think the downside is just the oddity of calling the resulting percentages "importances," because of course that's not what they are. But if your client likes that name and finds it useful, it doesn't cause any harm. Just exponentiate the levels of a given attribute, sum the exponentiated utilities and divide each exponentiated utility by the sum of the exponentiated utilities.
Say you have a 3-level variable with utilities of -1, 0, and 1. Exponentiated those are 0.368, 0.00 and 2.72 and the resulting shares would be 9%, 24.5% and 66.5%, and you could report those to your client. I would probably calculate these at the respondent level and then calculate the average across respondents.
I suppose you could think of these percentages as normalized odds ratios.
This is essentially a share simulation as if you were simulating as many alternatives as you had levels of an attribute, where all other attributes were held constant and the only difference between alternatives is the levels of the one attribute.