Gad, you guys use a lot of prime and throttle! I'd be afraid a
backfire would wreck the intake air filter. This is what I do:

First bow to the East and say two Hail Marys. Every aircraft is
different.

A trick I've learned (using mostly autofuel in a 172M) is to shut off
the fuel before shutdowns to empty the carb bowl - at least if it is
going to be sitting for a few days or more. Fresh fuel really helps
post-start running.
Be aware though that if the fuel shutoff isn't total, it will present a
treacheous safety problem if you somehow forget to turn the fuel on
again after a restart. It may, for example leak enough to allow
startup, taxi, and maybe even runup, but not support a takeoff. A
variation of this happened to me once with ice crystals in the
gascolator.

In a Minnesota/Wisconsin winter, I use 1 full prime preshot & hand prop
4 or preferably more blades before rolling out. I do the hand propping
with the mixture rich and the throttle completely closed. This helps
flash/vaporize the gas in the intake manifold due to the reduced
manifold pressure and the high air velocity across the idle port on the
throttle plate. Admittedly the engine may fire if some reason, the
mags don't ground out with the ignition switch. I don't have any
answer except to check for hot mags before shutting down earlier, and
to always be prepared for a hot prop.

Then I do a half shot prime during cranking and it gets me an immediate
lightup in one or two blades on my 172M. The second half of the prime
might be needed after a few seconds of running. Again, rank on a
nearly closed throttle (maybe 1/8 inch?) to help any fuel to flash in
the intake manifold. Three hands help.....

Adding taxi power for the first several minutes after starting with
autofuel causes a slug of fuel that has pooled in the manifold to go
thru the engine. When the engine is warm this won't happen.

Semi synthetic oil really helps. Any A/C with summer weight oil MUST
be preheated to get the oil viscosity within a safe operating range
before starting. Oil preheat on Lycomings, with the high camshaft
location, is really necessary. Even with the right oil, I get an oil
pump cavitation noise below about 32 deg F that tells me I'm pushing my
luck. Yes, the engine will start, but I suspect the post-start
lubrication is otherwise marginal, or worse, especially if it has been
sitting for a few days. Newer engines probably need preheat more than
old tired ones.

Take your time in taxi. The oil will be surprisingly slow to warm up
since the lub system of most engines has relatively little access to
the heat of the engine when the oil is cold and just blowing over the
relief valve. I know the manufacturers suggest otherwise, but they get
to sell rather than have to buy the repair parts.