Weird issue. When first starting out for the first drive of the day, when pressing the gas pedal, the vehicle requires significantly more pedal travel to accelerate. Once driven, acceleration appears to be normal. At first, I thought it was a dragging caliper, but this is not the case it seems, as it's generally only in the first few minutes of driving. It's also not temp dependent, as the vehicle is parked in a garage and generally about 40*-50*F and the times it's occurred the temps have varied to mild to cold. There doesn't appear to be startup rattle, and oil tested negative for coolant about 5k ago. Battery is original but currently does not have any errors.
Do you turn it on and go? For the first few minutes, even though the engine's at a fast idle, propulsion is completely by the hybrid system. You can tell by stepping on the gas during that time: the car will accelerate, but the engine speed does not increase. That maybe changes the feel? What were you trying to determine by that? If head gasket integrity, the usual failure mode is coolant leaking into combustion chamber(s). Have you ever cleaned the EGR system?
Oh, it'll bring in the engine if you step on the go pedal hard enough. There's normally a little delay, though, and if the engine is 16 years and 145k miles old and just cold-started in icy weather, it may be chuggy and boggy for a moment before adding much noticeable pull.
It's definitely draggy. You let off and it feels like the regen is cranked up like an EV, and when you press on the accel pedal it barely accelerates. We've had it for almost 15 years and both my wife and I independently mentioned it that we felt something was off. We are not flooring it but in order to accelerate now, you must really get into it. This wasn't the case before. And when it does accelerate, it does it at a slow rate. But again, only in the first few minutes of driving. Outside air temps are varied and vehicle is kept in a garage. EGR hasn't been cleaned, but again no issues until the past few weeks. If coolant gets in the combustion chamber it will go into the oil, which shows up in an oil analysis as sodium. The vehicle has slowly lost coolant over our entire time of ownership, needing refills every 20k or so of a cup or two, and I finally had enough and got oil tested. Not sure where it's going, but doesn't seem to be into the oil. It's done it since 40,000 miles.
Have you ever had hybrid battery replaced? All the symptoms you're pointing to sound to me like a hybrid battery that has lost capacity and needs to be reconditioned. Or alternatively your hybrid battery is being used alot going uphill before you get home and is discharged at the end of the previous drive. You could experiment by force charging the hybrid battery pack before driving to see if that addresses it. To do this you turn the car on, put emergency brake on and then put the car in drive while holding the brake down with your left foot and flooring the gas pedal. This will pump the energy the engine is creating into the electricity to charge the battery. When its full charged the computers will stop the process even if you keep holding the gas down. I'm willing to bet you're going to notice the problem is resolved. And if you upgrade to a Sodium-Ion battery pack the problem will be resolved even better than when the car was brand new.
That's not really true with the Prius.... By the time that happens, there will be no doubt you have a issue, and likely won't be able to drive it. If you are loosing coolant, you have a leak. It's that simple. Either the head gasket, or somewhere else. A simple pressure test will expose it. The heater core could be leaking, and you won't see it. Has anyone check the engine air filter? Cleaned the map sensor or other sensors? With the correct cleaner....
