Pretty much. If Pluto is a planet, then there are potentially thousands of objects that could be discovered over time that would then also be planets, plus updated models over the last century of the gravitational effects of, say, Ceres and Pluto, that showed that neither were capable of "dominating" their orbits for some sense of the word. So we (or the IAU, rather) couldn't maintain "there are nine planets" as a fact either way without grandfathering Pluto into the nine arbitrarily due to some kind of planetaceous vibes.

But the point is that millions of people were suddenly told that their long-held fact "the are nine planets, Pluto is one" was now wrong (per IAU definitions at least). And the reaction for many wasn't "huh, cool, maybe thousands you say?" it was quite vocal outrage. Much of which was humourously played up for laughs and likes, I know, but some people really did seem to take it personally.

I think the problem is we'd then have to include a high number of other objects further than Pluto and Eris, so it makes more sense to change the definition in a way 'planet' is a bit more exclusive.

Time to bring up a pet peeve of mine: we should change the definition of a moon. It's not right to call a 1km-wide rock orbiting millions of miles from Jupiter a moon.