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In my opinion, the historic center has more decent options than it seems. The trick is not to go during peak times and especially not to order randomly - lots of bars make decent spritzes but don't feature them prominently on the menu, so ask directly if they use fresh Aperol or Campari and if the wine is good. The difference between a mediocre spritz and a decent one often just comes down to that.
One thing nobody mentions is that some places around via del Corso have hidden spots in interior courtyards or upstairs - same owners, same quality but way less chaos than the street level. I've stumbled into situations like that by chance and the spritz cost normal price, not triple. Try asking if there's a quieter space before you sit at the bar counter with two hundred tourists.
And then it depends on what you're after - if you want the cheap classic version, then yeah, better to move around; but if you're looking for a place where the bartender isn't completely swamped with tourists and makes the cocktail without rushing, between the Pantheon and the side streets off Piazza della Rotonda there are spots that hold their own against the more famous hangouts.
Don't fall for the tourist trap bars around the Colosseum and Piazza Navona - they charge double and the spritz is mediocre. Try the more hidden areas like Campo de' Fiori or Trastevere instead, where you find places run by people who actually drink there and know what they're doing - the spritzes are fresh, the proportions are right, and the prices are fair. If you want to stay right in the center but skip the crowds, look for the alleys around the Pantheon - there are small taverns there that do an excellent job without trying to cash in on the location.
If you want a decent spritz in the historic center, forget the idea that you absolutely have to head out to the outlying areas - there are solid spots even around the Pantheon and Piazza della Rotonda where prices aren't completely insane and the quality doesn't suck. Sure, the typical tourist traps around the Colosseum are a ripoff, but not all central areas are the same; some bars that don't have much of a storefront make honest spritzes at reasonable prices without you having to trek to Trastevere every time you want a drink.
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