Been using Hario Mini Grinder for french press, and now for V60. Personally I think it's good enough.
I just used my French Press to make cold brew. I can make 100g coffee : 400ml water, which yield around 2xx ml of coffee concentrate

looks like cast iron burr. the burr design is similar to the ceramic burr where the burr sort of "crushes"(generates a lot of fines) as opposed to the sharpened steel burr that "cuts".
Maybe consider something like Timemore Chesnut grinder?  Easily available locally via Lazada/Shopee.

I recall Scott Rao getting pretty good extraction using a Timemore hand grinder in one of his instagram post.

Hi,
Will this be your first espresso machine?

What do you like about this machine?

So long as you manage your expectations accordingly.. There's isn't much you can expect from such machine.. You may get one which consistently gives a lower temperature extraction, hence the espresso is always towards sourish.. Or may get one that consistently have too high temperature and hence provides a bitter experience.. Stable and consistent temperature are hard to come by in such appliances due to its lower cost(small heating element, material used; e.g thin stainless steel vs solid brass)

We went through using similar machine before.. it was fun for the journey.. looking back, perhaps can keep the machine for milk tea.. pressurized and a deep/portafilter.. make sense for a quick Chatramue..

Brain fog because the espresso was too strong?

Let me guess... You tried the espresso using the "free beans" from the seller and you use your regular beans on your french press.. am I right?