Add the cost of the 10µF tantalum, compared to the cost of 1µF MLCC. It's some $0.10 extra, at qty=1000, just for the cap. PCB area also has a cost, so does having an additional line in BOM just to satisfy the regulator. Running outside the datasheet recommendations is really not recommended. I have tried to push my luck with 1117 and really got them oscillate. LM1117 datasheet specified minimum ESR of 300mOhm and minimum capacitance of 10µF, and these are listed as requirements. IME, better satisfy them.
For the question of "what is better, then", I already gave the answer: look at your distributor parametric search. The answer IMHO is anything that is stable with a small, cheap MLCC, say preferably 1µF, maybe 2.2µF. You likely have a 1µF MLCC in your BOM already, if you can use it (or put two in parallel), even better. If you have the regulator right next to the MCU for example, you can satisfy both regulator and the MCU bypassing by one ceramic. Fewer parts to place, easier design, easier assembly!
If it requires as much as 10µF, or if it requires a series resistor or using a tantalum, then I won't use it, unless it has some other magical feature, like extra low noise, very good PSRR at high frequencies, or very good regulation. Which 1117 obviously do not have.
For such sub $1 components, I look at the prices at 1000pcs or so. If you are just building 10pcs, the cost difference doesn't matter. But when you are looking at 100 suitable parts, you have to sort one way or another. When everything else is satisfied, I like to sort by price @ 1000pcs, that gives you an idea which parts are mass-market, and which are not. Mass market is good for availability, too! Cheap parts tend to have pin-compatible alternatives.
Chinese designs tend to use components where the local availability of drop-in replacement or counterfeit parts is good. The reason for using 1117 may be that the cost for them might be closer to $0.01, and local availability is good. If you buy from Digikey, just look at the Digikey prices.
Sorting by price, I find
AP2138 for $0.12 at qty=1000, and this is stable with a 1µF ceramic (min ESR is 10mOhm). But I didn't carefully look all the specs to see if it fits the OP's question. It satisfies 200mA current at least. The package is nice, small, yet easy to handle, but not very good at thermal dissipation so do the math.
Having Vin(max) = 6.0V really widens your options. This brings you the cheap, modern selection made with the same CMOS process most modern ICs are made, some in very small packages. These parts are designed for things running at 5V, or from a single li-ion cell, have abs.max. somewhere around 7V and recommended max around 6V, and are fine if you have a regulated 5V bus available, with the usual filtering and damping in place.