Sizing transistors for linear operation will become very expensive. Because with higher voltage than about 3..4V across the FET, the allowed current droops very rapidly, usually ~1/Vds^2. The problem is, the FETs do suffer from thermal instability over their silicon area in a similar manner as bipolars do. Claims like "FETS do not suffer from second breakdown" is not true anymore, once the FETs reached the low Ron and so the high current capabilities. It is displayed in the SOA chart...

With the pulsing it is worse than a single pulse, but still tolerable. The point is that way it allows to use transistor way smaller than would be needed to drive the full cold lamp current (about 14x the nominal), yet stay within the safe operating area of the transistors. Mainly when integrated into some more complex IC (one chip driving a bunch of bulbs and so on, where the silicon area (needed fot brutt force power) is way more expensive than discrete transistors, but the density of small transistors allow to integrate way more complex control into fraction of the power element on the chip. Or whenlarge currents are involved, so a microcontroller alowing the more complex scheme is cheaper than the brute force power transistor sizing.

Such warmup burst uses to be some dozen of pulses, so not that much stress from the battery ripple. But having similar thing present all the time could wear the input connections (reverse battery protections and the input capacitors in many of the electronic modules). Plus it may affect many sensors, temporary burst of inaccurate data from the sensors could be ignored (the car electronics count on such starting bursts happening from time to time), but for longer time would cause problems (may trigger fault codes, so they should not las for so long).

The alternator has no problem at all, as it won't respond that quickly anyway, such surges are handled solely by the battery. The alternator only takes over the average current, just maintaining the battery charge level balance...

Using high frequency DCDC means way more complicated and demanding power circuit, you need to put all the smoothing within the control box. Way too much hassle for something as slow as an incandescent.

Where do you came to the 10V?

We were talking soft-start by gradually opening the MOSFET here, right? So it will be subject to circa 14V ~20A for a few hundreds of milliseconds, with a voltage gradually collapsing to zero.

To keep the discussion sane. Everyone imagine connecting one of those cheapo 'modified sinewave' inverters to the car power, let's say in 500-1kW. As those actually generate the voltage in +/0/-/0/+/ manner, the problem is the same as with the circuit discussed, heavy squarewave input current ripple at 100/120Hz. May be just marginally smoothed by the transformer stray inductance and some small electrolytics at the input. I have not heard of much problems these units cause to the on-board 12V power.

Making a smooth DCDC buck converter to power the lamps at 12V 500W is actually not that complex, just two MOSFETs and inductor plus some low-ESR electrolytics, and in fact a suitable inductor can be easily borrowed from some surplus ~500W 12V AC power supply, there also may be some high current MOSFETs in the synchronous rectifier part. And even a PCB chunk with these and the inductor could be conveniently re-used! One should have some experience in power electronics, though, I am not afraid, but playing with this may be slightly off scope for an average electronics amateur.

@"10V": I did not get the link to the linear operation, that makes sense.

Designing a DCDC  is possible, but it is veverything but a trivial thing to make it reliably working. It is about managing all the parasitic in the physical construction, all getting even more complicated when high currents are involved. Pretty complex for someone "fluent" in such high power high frequency stuff. So I consider that far out of the reach here.
I think very good indication could be you will never find anything like that in any commercially made automotive equipment, but pretty much all you find is either hard brutt force switching of the incandescents with a driver capable to directly handle the cold current, or using the burst mode. And if dimming is required, it uses with a low frequency PWM, never any high frequency DCDC. I would guess there would be a reason...

For the ripple you are right very likely the vehicle can handle it, in fact normal 65W halogens are dimmed like that to compensate for the charging voltage fluctuations in modern cars (the modern battery management sometimes drives battery close to 16V, without dimming that would kill most incandescents in a very short time). But there the left and right sides are synchronized so to minimize the overall ripple.
It is not the vehicle is not able to handle the ripple from time to time, it is just it creates extra stress.

The fact those inverters do not destroy things at once does not mean they are healthy for the car. Plus a topology using chopper at the 12V side and then a step up transformer at the generated mains frequency is used only with really the worst products.

Normally such inverters first use DCDC to convert the 12V to about 330V (for a "230V" output), use rather large capacitor tank there (as that is way more compact in size for the given energy storage than what would need to be at 12V) and then direct, slow switching (at the 50Hz) inverter bridge generating the MSW from it. With this all the ripples are handled by the tank capacitor at the 330V, the 12V is just providing the smooth power for it. So then you have very small ripple on the 12V side even with rather heavy loading at the AC side.
I don't think it is even possible to make an inverter using low voltage conversion to AC at 1kW level, it would require about 200A connection from the battery.

Plus very important, these invertors are never loaded anywhere near full power for any significant time, it is just for their ability to handle surges from various things (like fridge compressor,...). The running power uses to be in 100W ballpark, not much more.
Of course, when someone uses it with a 300+ Ah battery bank, then the situation is different, then these batteries won't have any trouble delivering the 20A ripple and still keep the voltage ripple on the battery line reasonable. But I guessed we are talking about standard battery in the car, so nothing that low impedance.

The fact here it will happen only in case of one bulb failure means the accumulated stress won't be that much, so unless it is expected to operate with blown bulb for months or years, the ripple won't be that big deal. In that I was a bit conservative, as normally similar long term pulsing does not exceed 5..10 A maximum (lighting dimming, ignition, fuel injectors), but here we are talking about 20A nominal, normally there is nothing long term pulsing with such high current (inrush current surges from things like power steering, ABS, or some modern transmission hydraulic pumps I do not count, as these are rather short burst with quite long gaps inbetween).

I have a 400watt 120V square wave inverter... should I stop using it and replace it with a Pure sine wave version?

Depends how you are using it and how it is constructed inside.
If you are using it for a short time here and there, I don't think there would be any problem.

If it is the heavy low frequency transformer design, I wouldn't load it by more than 80..100VA static load (and yes, I wrote VA, because it is the apparent and not the real power, what translates into the input ripple loading). Not only because of the input wire loading, but because these things tend to not handle more than about 1/4 of their rated power well for longer time, they tend to overheat.

If it is the more modern type with the 14->170 V (for the "120V" output) DCDC and a bridge to generate the MSW, I don't think there would be any problem, the 12V side is really handling just a smooth power, all the ripple stays at the 170V siode between the transistor bridge and the tank capacitor.

But the main problem with MSW (vs true sinewave) is not that much how it behaves towards the 12V side, but how it stresses what is connected to it on the 120V side. Many devices are just not handling the steep MSW edges well, regardless if they are behind a bulky transformer, or just generated by a transistor quad. And with many modern loads the inverter will suffer a lot, again regardless whether it is the old crude "iron brick" or the more modern design with DCDC and HV inverter.
So if you are considering any upgrade or refresh of an old equipment, so looking to buy some new one, or equiping a new car with 120V power, I would definitely go for true sinewave

I have a 400watt 120V square wave inverter... should I stop using it and replace it with a Pure sine wave version?

Why bother? If it works ok for your purpose, don't fix it.