Does it make sense to buy [a] Speedlight?
Absolutely if what you want to learn and achieve is good on-camera bounce flash. That is exactly what a speedlight with a head that tilts and swivels is designed to do that a pop-up flash cannot.
However, very few of us would think a pop-up flash is "too strong". It's likely that you still need to learn that flash exposure is controlled by iso, aperture, power, and distance. Your subject may have been too close, your ISO too high, or your aperture too wide, and that caused the flash overexposure. Speedlights are actually more powerful than your pop-up, while still being on the low power-output end of the spectrum when it comes to lighting gear.
Will it cover most everyday situations where I would need to shoot children photos (birthdays, portraits, children playing etc.) or would you recommend some other solutions for me?
Again, absolutely. I've never felt a need to go beyond speedlights. And it's the ideal solution for you as a beginner. A hotshoe flash is unique among lighting tools in that it can be used both on-camera and off-camera and being battery-powered, it's very portable.
I am not looking to set up a home studio, so heavy equipment is not my thing but something light and versatile will be perfect.
The magic here is that you can also use speedlights off-camera, Strobist-style to set up a home studio, and still have something light and versatile and portable.
So which one would you recommend? Is there some best buy option where intensity of light can be adjusted?
Absolutely. But the models I recommend today in 2021 (Godox TT685 and TT350) are not the models I would have recommended even six years ago, and they are likely to be replaced and superseded as models in a year or two from now, being 5-6 year old models today. There are reasons we don't like shopping questions on Stackexchange.
So, instead, I point you to this Q&A: What features should one look for when selecting a flash?, where you can put together a list of features you think you'll need and then match them up against existing current models. Because, like lenses, which one you need is up to you and how you shoot and personal preferences, as well as budget.
Also, a recommendation for a beginner tutorial on how to use it would be great.
For on-camera bounce flash, I always recommend Neil van Niekerk's Tangents website. He's a professional wedding and portrait photographer and he's a wizard with an on-camera speedlight. He can also teach you off-camera flash, but the main go-to source on the web for that is David Hobby's Strobist website. But. Learn to walk (on-camera bounce) before you run (off-camera flash). And, if you don't know how to stand yet (exposure triangle mastered; comfortable in M mode), that must be solid before you can start learning flash. Because while ambient-only exposure is juggling three balls (iso, aperture, shutter speed), flash+ambient exposure is juggling five (iso, aperture, shutter speed, power, and distance), while riding a unicycle (balancing flash against ambient). :)