In my Shul, most Minyanim are kept relatively quiet during Davening when strictly required, which generally means from Baruch Sheamar through the end of the silent Shemoneh Esrei. That is not to say there aren't some problems, and also some talking later in the Davening when inappropriate even if not strictly forbidden, but generally it is under control. So to say "no one"..."follow the Halacha" is simply not true.
That being said, I see two particular problems, to varying degrees:
1 - Due to the relative leniency during Pesukei Dezimra, there is a tendency for some people to talk more than they should. For example, during Pesukei Dezimra there are more permitted responses - e.g., Amen to almost any Bracha - than there are between Borchu and the end of Shemoneh Esrei. Confusion about what is and is not permitted, even when the vast majority of what is permitted is all Tefilah-related, and not the stock market reports or sports scores, lends a certain informality to Pesukei Dezimra that can be conducive to talking. In addition, the contents of Pesukei Dezimra, while not optional, can be cut short in order to get to Shemoneh Esrei with the Tzibbur, which is another factor that can make it seem like Pesukei Dezimra is somehow less important than the rest of Davening, and therefore "OK to talk".
2 - I was concerned, and my concerns were quite warranted, that the current COVID-19 situation would lead to some significant problems. Specifically, in my area, as in many areas around the world, Davening basically starts with Borchu. On weekdays everyone except the Shaliach Tzibbur says through Yishtabach at home and then goes to Shul, and the Shaliach Tzibbur says up to Yishtabach at home and starts with Yishtabach in Shul. Shabbos is slightly different, and I am sure there are variations in some places. But essentially Pesukei Dezimra is said at home. This can enhance Pesukei Dezimra, as there is no rush to keep up with the Tzibbur, no need to skip anything, etc. However, the journey to Shul poses a problem. At least a couple of times a week, I am greeted by people on the way to Shul. As I understand it, a simple greeting is actually permitted Halachically - and to not respond in a friendly way would be insulting and problematic. However, often the greeting extends to discussions of various sorts - typically about relatively important matters (e.g., this past Shabbos someone proceeded to inform me of a recent COVID-19 case in the neighborhood, which I had not been aware of) but almost always not immediately relevant (not about the Davening of that morning) and therefore really not proper to discuss either during Pesukei Dezimra or between Yishtabach and Borchu. It is very hard to give a Halacha lesson to someone honestly being friendly, often much older than I am (and therefore deserving of respect), etc. This is likely not the type of talking referred to in the question, but is still a Halachic concern.