I think the ideal use for the home planet is to start off as a manufacturing planet. Ultimately, I plan to make it a manufacturing/military planet - which is to say that all the military 'hyperion' buildings will go there as well to gain the maximum value in terms of bonuses & usefulness. That way, all the ships built on my home planet will have all the hyperion advantages. With the hyperion tiles, the adjacency bonuses are important.
Also, I colonize my in-system second planet (usually Mars) quickly, and make it a research planet, because you need research immediately. For people with production of >1 capacity, It's easy to create a second shipyard on the adjacent planet to facilitate this production using the home world's extra capacity, without any distance hits.
I usually go for the terrain-enhancement techs to add to my planets, including the home world. To call the home world a production is a bit of an oversimplification, so the enhancement tiles are quite necessary.
For the production/military planet to be at maximum you will need to install 'farming' tiles, and hospital tiles, and possibly 'happiness' tiles as well, though I prefer to solve the happiness issue with starbases. Farming tiles are necessary to get the population up, and hospital tiles are needed to rebuild the population after sending out colony ships and transports. Don't remember if freighters require people or not. Colony ships & transports drain pop quickly. I may even retain a transport to shuttle extra people from my in-system planet into my home planet.
Overproduction is usually not an issue for me on the home planet. There are plenty of civilian tiles to build on the planet, so this is really a matter of adjusting the manufacturing slider to accommodate both needs,even if it is occasionally on a turn-by-turn basis.
Making the in-system planet a research planet is just a stop-gap for research until my expansion kicks in. Once I have lots of research planets, I usually switch over the in-system planet to production with a shipyard. This 2nd production in-system planet allows starbases to benefit from having two production planets in the same system. This second shipyard I usually dedicate to making constructors and, occasionally, transports - to ease the burden on the home world.