Probably not.
If I was signed to Fender I would have used a Fender for both the studio recording and live promotional performances of the song. If the Jazz bass Hamilton is playing in the YouTube clip is good enough for the tour promoting the song then it should have been good enough for the studio work recording the song. Or from the flip perspective, if a Thunderbird was good enough for the studio work recording the song then a Thunderbird should have been good enough for the live tour promoting the song.
Again, I suspect (but have no certain knowledge) that Hamilton was signed with Fender and probably had a clause in his contract at the time that required stage and live appearances to be Fender only. If that assumption is correct, it potentially says a great deal about his opinion (and mine as well) of the Humbucker equipped Thunderbird being a much better fit for that particular song than a couple of Jazz single coils or a split P pup.
I've only ever played one Fender that can hang note for note with the Thunderbird, the Fender Telecaster Modern Player which is also equipped with Humbuckers. Fortunately I bought mine before they were discontinued. If you ever see one for sale for less than $600 snag it! It's the Eppi VP of the Fender world - and it doesn't even have Tort as an option.
Here's a steal...
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