The Shulchan Aruch 59:4 says not to answer amen after Ahava Rabbah/Olam.
The Rema to 61:3 says that the custom is people finish Ahava Rabbah before the Shatz, so they can answer Amen.
The Mishnah Berurah 59:4 § 25 (linked above) explains that Ahava Rabbah isn't a regular birkas hamitzvah, where you shouldn't speak before doing the mitzvah. It sounds like from him that the Shulchan Aruch disagrees.
The Mishnah Berurah writes that the Achronim agree with the Rema that you should answer amen if you finish before the shatz. However, he finishes by writing:
אך לכתחלה יותר טוב שיסיים בשוה עם הש"ץ ולא יצטרך לענות אמן אחריו
The practice you've observed is based off this Mishnah Berurah. He advises to finish along at the same time as the shatz, so as to not be obligated to say amen, seemingly to avoid this dispute.