Alright, let's pick this apart:
Doesn't require an initial bloom
Blooming isn't needed in immersion brewing styles. You do it in filter brews so all of the grounds get wet as soon as possible, which helps with an even extraction. When you (presumably) dump all of the water in your french press at once, there won't be any non-wet grounds inside anyway.
Doesn't require any timing
Timing is done for one thing and one thing only: consistency. Some coffees, especially lighter roasts, can be a little finicky to extract. Being able to get a good tasting coffee each time consistently is greatly helped by timing your recipe and doing exactly the same thing each time.
Put 60g in the container. Pour in 1L boiling water.
Pretty standard ratio for most french press, Aeropress and filter recipes.
When the grinds float to the top, stir them down.
While this doesn't greatly improve extraction, it prevents floating grounds from clogging up the mesh or landing in your cup when pouring, so I'm all for it.
Press.
Have you tried not to? I mean why press? If you don't like grounds ending up in your cup and the brew tasting sandy, then don't press the plunger down, simply put it in the french press and use it as a strainer. When you press it down, you agitate the water and also help suspend more grounds into the brew, which means more grounds end up in your cup when you pour it. You may like it, and in that case keep doing it, but I don't really like sandy coffee so I'd avoid it.
Stirring a few times ensures all the grounds are equally soaked.
Ideally, all of your grounds get soaked when you pour in the water, stirring more than once or twice doesn't really help that much with extraction, but it does increase agitation, and thus you get a grittier brew.
In conclusion: if your coffee makes you happy the way it is, don't change anything, you're the one who has to like it. The only thing I'd recommend trying is not pressing the plunger down and timing your recipe so you can get the perfect brew, exactly how you like it, every time. You say you use 60g of coffee per 1l of water - if you're weighing your coffee anyway, perhaps you can buy one of those scales with integrated timers, so you don't have to use a separate device for it.