Spellwrought tattoos are similar to the spell scrolls but extremely more powerful.
There's no restriction to casting time, while a spell scroll needs an action to be cast. It doesn't require an action to draw it from your bag or at least doesn't require a free hand to use this. There's no limit to how many you can have in your body, you can potentially have infinite spellwrought tattoos.
Is there some reason to this item to exist? Does this really fits in the game?
You can have a fighter with many shield tattos that gives +5 to his effective AC.
Or you can many absorb elements tattoo and be resistant to most of the kinds of damage in game. With common items which are indestructible. Common items by the magic item price rule of Xanathar's guide to everything has it's price at (1d6 + 1) x 5 gp*.
Casting Time
Spell scrolls has it's cast time bounded by the item. You can't use a spell scroll as a reaction or bonus action because you need an action to read it. What doesn't happens to the Tattoo, which is only limited by the casting time of the spell.
Requiring a hand to use
If you are using a spellscroll of a spell wich doesn't require somatic component, you will still need to have a free hand to cast it as long you need to read it, so you need to have it positioned in front of you, and usually you use your hand to this despite there are other ways to do the same. When you do the same with a tattoo, you casts the spell as if you had it prepared.
Limitation to number held
There is already a tweet from Jeremy Crawford about the tattoos where he says you can overlay a tattoo to another without loss.
https://twitter.com/jeremyecrawford/status/1331691854832701440
Repeated usage
Spellwroought tattos are consumable items its base price is halved. So it costs, at mean, 22.5 gp to have access to any first or second level spell of any class. Many of that spells as shield and absorb elements scales wonderfully until the late game. And high level campaigns has a lot of money.
Class and level bounding
Spell scrolls are class and level bounded. You can't cast a spell with a spell scroll if it isn't available to your class. It is also bounded to your magic casting level. You will need to be successful on a DC roll if you try to use a spell scroll which has a spell of a spell slot you don't have access to.
Tattoos don't has any of these limits, the only it has is there's no tattoo for spells from above the fifth level.
Best Answer
There are multiple restrictions on the Spellwrought Tattoo
The restrictions require a little bit of cross-referencing with the Dungeon Master's Guide and inference, but they certainly exist. To go through your objections in order:
Casting Time
Since the tattoo does not provide a casting time, we refer to the default rules for how magic items grant spellcasting. From the Dungeon Master's Guide: "The spell uses its normal casting time, range, and duration, and the user of the item must concentrate if the spell requires concentration."
Note that this is identical to a spell scroll, which explicitly "requires the spell’s normal casting time".
Requiring a hand to use
While the default rules for spellcasting remove the need for any components, the tattoo specifies that its casting only omits material components. We can read this to override the general rules for magic items, and infer that casting this spell does require somatic components, which will typically need a free hand.
Limitations to number held
You'll notice in the description of the item, area covered is linked to rarity of the tattoo (and thus the spell level available). If you already have a tattoo covering a limb, you presumably cannot have another tattoo on that limb (since there would be no room).
As linked in the question, Jeremy Crawford has stated that this is not the case. However, his tweets are not considered official rulings, and I suspect many DMs would rule to the contrary here.
Repeated usage
And of course, when you use a tattoo, it goes away. Paying 100 GP for a one-round boost of 5 AC is... perhaps not the best deal, depending upon the availability of gold in your campaign.
Isn't this still better than the equivalent spell scroll?
Well, yes. There are many magic items that are more powerful than other items of the same rarity. At the very least, being able to use these without being of a class that could normally cast the spell is very strong utility. 5th edition doesn't try particularly hard to exactly balance its items against one another, relying on the DM to distribute them at a cost and frequency that makes sense, or to correct if things go too far.
So while it's theoretically possible that you could pour all your money into unlimited uses of shield, it requires your DM making that tattoo available, your DM not changing its price to account for its increased utility over other common consumable items, and your DM ruling that you can overlap multiple tattoos over each other.
Spellwrought tattoos are not broken, they're just better in a lot of scenarios. That's allowed.