There's a difference between a rushed process and a shortened process. The reason for these phases not operating in parallel is not so much about what is best, but about what usually is most practical and fits with capacity.Which is shortening the process, something the poster you quoted disagreed with yet you somehow seem to be agreeing.
If running these phases in parallel was the best way to do it then it would have been the normal process in the first place. It wasn't.
The process of developing a covid-19 vaccine has been unique in several ways. There are more companies working on it, and high infection rates meaning you can more qucikly know if the vaccine works: researchers have been able to 'follow the wave' with trials, recruiting participants in different countries as new spikes mean infeciton rates are sufficient to test in those countires. Governments and organisations have been preordering unproven vaccines, meaning companies could start on the manufacture of them: meaning there's no delay between approval and roll out. There was also the advantage of being able to use the work already done on producing SARS-1/MERS vaccines: which had stopped because these variants of coronavirus proved less robust in various ways and didn't spread sufficiently for the demand for a vaccine to appear.
Maybe about Greggs you have a point regarding short amount of time being inside but certainly not McDonalds/Burger King. You wait for your food, you can still sit in and eat your food(that may not apply to them all and admittedly I have not been in one since lockdown so that may not be the case now) and alot of the customers are teens who we all know a large proportion don't wear masks so could easily be spreaders of the disease.
With regards to take-outs, this paper is quite helpful in describing different risks in relation to distancing time spent next to someone. Even withhout masks, short-period indoor contact in well ventilated settings is described as low risk. A good sized, modern McDonalds where you're in-an-out in 5 minutes, or a small Greggs where the doors are open, shouldn't be considered a high risk setting. In relation to public transport, btw, the writers note the low evidence for spread on airplanes. We could apply this sort of analysis to metro/tube type rail services and see that these should be considered very low risk sites too.