Prius absolutely is full hybrid, meaning it could go decent speeds with the engine off. All the way back to December 1997 in Japan, and of course 2001 here in America. The Prius that was just discontinued to make way for the fifth generation in 2023 goes at least 55 if not 65 with the engine off, and these are not even plug-in models.
So Prius second generation 2004 through 2009 has, in the United States models not Japanese models, a thermos behind the front left corner of the bumper. It holds probably a gallon of coolant, and I have personally experienced it keep that coolant hot enough to burn skin after six hours sitting in 40°F Outside . That being said, no, the car really doesn’t struggle to stay warm. Although, the HVAC computer calling for heat might make the engine run more than it should otherwise. But, what you were talking about, literally cold starts, typically only once a day. If your Prius sits outside all day when you are at work and it’s winter, it might have a decently cold start when you go to leave to go home, but that thermos injects a decent amount of heat into the motor
Generation three and four, and I am assuming generation five, have a coolant sleeve around the catalytic converter which superheat coolant quite fast on demand. They truly didn’t think of everything. It is amazing to think that 25 years ago, there were cars doing 40 with the engine off, but weren’t restricted to a plug. I am not a fan of electric cars, for a lifestyle like mine and my families, they are an absolute scam. We are adventurous and often go places where an electric car would slow us down because it would need to be charged on 110. A hybrid is the best of both worlds. The only reason we don’t have the new Toyota sienna hybrid is because we are not spending $50,000 to replace our perfectly good 200,000 mile odyssey