Antennas are not amplifiers. As long as the antenna is correctly tuned to the impedance of the transmitter output stage, it will radiate most of the power of the transmitter, but cannot exceed 100% (it is a passive element, and exceeding 100% would break the laws of conservation of energy). In this case the size of the antenna does not affect the total power emitted and there will be no range gains from using a bigger antenna.
What the antenna size/shape can affect is the directionality of the beam. By emitting the same power directionally, you can increase the range in some directions at the expense of reducing the range in all other directions. But the total power would be still the same.
Also the reception gain is actually the same thing as emission gain. You cannot make a passive antenna that has better reception without making it more directional, which means worse reception from some other directions. You cannot improve reception from all directions by making a bigger antenna.
Of course a bigger antenna can pick up more power. Just imagine placing two antennas instead of one. Each one receives a certain amount of power. Two will receive the sum of that power.
That doesn't work that way. If you sum the signals from two antennas, those signals would differ in phase depending on the direction of the incoming wave. For some directions they will add up, for others they will cancel out. That would change the directionality of such antenna system, which is exactly what I have written above - you gain somewhere, but lose elsewhere. Antenna gain is a function of only its directionality. A theoretical, ideally symmetric antenna has gain 0 dBi, regardless of its size.
But then it is no longer a passive element. Sure, you can use N antennas connected to some electronics that adjust the phases dynamically so they always add up. This is what modern "beam-forming" wifi routers do these days to improve range.
Bigger antennas can positively pick up more signal. The Mars rovers do not have megawatt radio transmitters, instead we use enormous antennas on Earth.
We use antennas which are incredibly directional for Mars rovers. Compare a regular lightbulb vs a flashlight with a lens. Total light emitted might not be that different, but the intensity at the same distance (or how far the light reaches) is because the beam is focused. That works the same with receiving signal: a giant dish will focus the signal to a central point, and a Yagi-Uda antenna will use a similar phenomenom to focus the signal on a dipole.
Ofcourse car bluetooth presumably has an antenna that radiates roughly sphere-like, it wouldn’t make sense to pick up signals on the right of the car with a dish or something.
To that end, bigger antenna doesn’t do a whole lot to boost signal. There might be other factors at play increasing the reception though, such as signal polarity and whether the length of the antenna matches the frequency in a correct way (or is some multiplier of it). For Bluetooth that’d be roughly 13cm or 6.5 or so.
Because it wouldn't be as directional as a big dish. The relative size of the dish vs wavelength plays a factor here. If you make a dish thats the size of the wavelength it wouldn't even work as a dish, it would receive the signal from all angles. This is also why high gain antennas for higher frequencies can be much smaller than for e.g long waves.
It works two ways: one is that you can go lower in frequency (e.g. bigger wavelength) if your dish is bigger, the other is that you just capture more of the signal. If you take light as a metaphor, a bigger lens will just provide more 'signal' at the focal point.
What the antenna size/shape can affect is the directionality of the beam. By emitting the same power directionally, you can increase the range in some directions at the expense of reducing the range in all other directions. But the total power would be still the same.
Also the reception gain is actually the same thing as emission gain. You cannot make a passive antenna that has better reception without making it more directional, which means worse reception from some other directions. You cannot improve reception from all directions by making a bigger antenna.