Okay, let's take a few seconds and reflect on what you need to establish the flame and what you've actually got.
A.What you need:
1. You need gas at the proper pressure (3.5 IWC in the manifold whenever the main gas valve is open.
2. You need a solid pilot flame to ignite the gas when the main valve opens.
3. You need good flame propagation from the pilot to the first and subsequent burners (this needs to be almost instantaneous).
B. What you've got:
1. Gas at 3.5 IWC in the manifold.
2. Solid gas flame.
3. Light-off but not smooth and sometimes the pilot blows out.
From a comparison of what you need versus what you've got, I'd say flame propagation is your issue. Remember that water is a by-product of combustion and water creates rust. Rust not only deteriorates burners over time but also clogs the burner outlets.
Really old burners were made of cast iron with a lot of little holes for the gas. These units lasted many decades as cast iron resists rust quite well. Newer atmospheric burners switched to stamped steel tubes with louver type gas outlets. These louver type tubes can get very easily plugged with soot and I've seen many where the louvers were damaged or missing, thus allowing too much gas to enter the flame area.
Finally, when you advance to the fan assisted combustion, you get to the in-shot burners with their airplane like wings that are the crossover tubes. Those crossover tubes must also be kept clear of any rust and/or damage that would close them down. Cleaning crossover tubes should be part of any annual service but many techs don't even know they're present.
In summary, I believe your issue is a result of slow flame propagation leading to periodic sudden over pressures as the unignited gas accumulates and then ignites.