It seems to me most SE strategies emphasise "cottaging" your capital, and leaving it as the only commerce-generating city, rather than converting it to a food/specialist heaven like all other cities in SE (except the production ones).

I have a question how vital is it for the strategy, i.e. whether it greatly hinders you if instead you focus on food and specialists in your capital as well?

I understand having high commerce production in your capital is useful if you run Bureaucracy, but is there any other benefit of that tactics?

It seems to me most SE strategies emphasise "cottaging" your capital, and leaving it as the only commerce-generating city, rather than converting it to a food/specialist heaven like all other cities in SE (except the production ones).

I have a question how vital is it for the strategy, i.e. whether it greatly hinders you if instead you focus on food and specialists in your capital as well?

I understand having high commerce production in your capital is useful if you run Bureaucracy, but is there any other benefit of that tactics?

Some people think cottaging the capital is a good SE strategy, but I don't think that's pure SE, well unless late in the game when you capture your enemy city with mature cottages all around. High food output means you can run more scientists if you run Caste System. You can mine the hills and eventually be able to build workshops and watermills/lumbermills, which you can benefit from running Bereaucracy as well.

If I have some cottageable terrain in the capital, I'll cottage it. However, I also often farm up my capital if it has a lotta food and then I'll just try and put the GL there with 6-10 scientists on top of that. With pacificism you can crank out a LOT of GSs that way.

I have a question how vital is it for the strategy, i.e. whether it greatly hinders you if instead you focus on food and specialists in your capital as well?
It isn't vital but it is often a good idea. You have to get money from some source and the capital is often a good place to look. But, like everything else, it depends on the terrain. If the capital has lots of freshwater, it is often good to build farms instead and just find some other place to have a commerce center.

One viable technique is to invade your neighbors and use the gold from pillaging and capturing their cities to fund your maintenance. That has the drawback that eventually you might run out of enemies for a period of time -- say, on a continents map prior to Astronomy -- but I have used this strategy successfully to jump-start my early game. Bureaucracy is a useful civic, but in this scenario Vassalage or Nationhood might be better, in which case having a cottaged capital isn't as much of a benefit.

Generally it very depends on dificulty level and isolated or no isolated start.
On easy level one quickly run out of trade partners, on isolated start one does not have trade partners.
In this case on need to self research a lot of non ligthbulbable tech. In this case commercial capital is advisable, otherwize it does not go well with useal SE or FE strat.

It seems to me most SE strategies emphasise "cottaging" your capital, and leaving it as the only commerce-generating city, rather than converting it to a food/specialist heaven like all other cities in SE (except the production ones).

I have a question how vital is it for the strategy, i.e. whether it greatly hinders you if instead you focus on food and specialists in your capital as well?

I understand having high commerce production in your capital is useful if you run Bureaucracy, but is there any other benefit of that tactics?

I think it depends on the type of SE you chose to run and that depends to some extent on your leader traits. For instance I will cottage up the capital in a game when using Alexander (Aggresive and Philosophical), particularly if I do not intend to run Caste System but will be whipping out a lot of troops with Slavery for most of the game. My capital will then be my Science City as well, getting the Great Library and Oxford eventually, so cottages and Bureaucracy will deliver a lot beakers as long the research slider is kept high.

In this situation building all farms on grasslands and floodplains in the capital would make too much food and require the Caste System to be run ... and that conflicts with the ability to make a powerful army fast (in other cities) and to incorporate the captured cities into the empire. Getting and spreading one religion and running Pacifism makes Alexander under Slavery and Hereditary Rule a very aggressive version of SE. The key early techs are Currency and Code of Laws and running a few scientists and merchants from libraries and markets (each giving 9 GPPs / turn) can make a huge impact on winning the Liberalism race and afterwards, while financing an expanding empire.