Companies selling insulation are going to say that the insulation they're selling is great in every way. They're not a reliable source.
Sources are the issue - there are some inside your home. Everything powered and everything alive. If your home was uninhabited and everything was turned off 24/7 then insulation might reduce the temperature inside in summer. Or it might not, if you take the whole day/night cycle into account. Insulation would slow the rate at which the interior heated up when it was hotter outside than inside but it would also slow the rate at which the interior cooled down when it was hotter inside than outside. So my guess is that it would be cooler in the morning and warmer in the evening, on the whole.
I had a boatload of insulation added to my house a few years back. It already had cavity wall insulation and loft insulation, but I had a load more loft insulation added and inches of insulation added to the surface of the walls. I lost some space in the rooms at the front, about 10cm maybe, but I had more space than I needed anyway. It's on the outside at the back. It's definitely better in winter, but overall I think it's a bit worse in summer. I'd say it's the time-shifting I expected - cooler in the late night and early morning, warmer in the evening and early night. But I'd say that it's an improvement overall, with the advantage in colder weather outweighing the disadvantage in hotter weather.
In a well insulated house, the most effective response to hot weather that I've found is to sleep downstairs. There's a significant difference in temperature between upstairs and downstairs in my experience.