PT has its place in the beginning to tell you not just what to do, but also HOW to do it, which is as if not more important than simply what to do. It also tells you what NOT to, at least for now.
Keep in mind that “core strength” is way way beyond sit-ups and crunches or oblique work, whether standing up or lying down
Core is armpits to hips, 360*. If you don’t have hip mobility and flexibility, your core suffers. If your shoulders have limited ROM, your core suffers. If your foot or ankle stability is weak, your core suffers.
I have been getting stronger, more mobility in my hips, more stability in everything, for 20 years, at home, than I ever did in the previous 15 years at a gym, and with less equipment but with way better programs actually designed to improve all those things.
So yes, you can do it.
What prompted the “you need a stronger core” Rx? That matters the most, in terms of where you should start.