If I'm in a band doing original material as the primary, I want to redo a cover in a style recognizable with our band. Like Clapton did with Dylan's songs like "Knocking on Heaven's Door" or what Hendrix did with All Along the Watchtower.
If I'm in a Top 40 / oldies / R&R / country/ funk / R&B dance band like I did for .....too long........I made it a challenge to come with tones matching the original and I hope I learned something about sound in the process. So you take all your favorite tones and phrasings and the total of what you learned becomes your style. I think it helps your confidence as a player to be aware of that, other than feeling like a copy artist.
Current technology modeling amps make coping the tones from the original recordings a hell of a lot easier than a one trick pony amp, as good as that amp might be. I think that's what modeling amps are best suited for, as long as they have the watts and the speaker(s) to deliver.