Any real manufacturer, even cheapest Chinese ones, have absolutely no issues with 0402 or 0201. I think advice to stay on 0603 is from 2010 or so and considers lowest-quality random fabs of that time.
If there is a chance you would use a $1000 DIY P&P kit, or use as a CM some tiny company which uses something like that, or if you hand-assemble a lot, then limiting to 0603 might be a good idea. Otherwise than that, I find that forcing one's design into 0603 has negative influence on ease of design and board size around large pin count devices such as microcontrollers. Difference in part or assembly cost is negligible, but your design ends up larger, or if you have size constraints, you spend more time routing the PCB, placing bypass caps and series resistors further away from ICs just so you can fan-out the signals, and for what - because you read on forums to avoid 0402.
Really, only reason to artificially limit yourself to 30-year old technology node is if you hand-assemble a lot and have troubles with smaller parts.
Then again, sometimes it doesn't matter. For example all my 0.1-1uF generic bypass caps and 33R series termination resistors and 4.7k pull-ups and similar are in 0402, because many of them often go in tight spots close to ICs, but then if I design e.g. an amplifier circuit or voltage regulator feedback circuit might as well use 0603, doesn't usually matter at all. 0402 is in pretty sweet spot because you can still quite easily hand-assemble prototypes (it's not as nice as 0603, but doable and that's enough for prototypes), yet it's small enough even in real production so that you don't start feeling like you are wasting a lot of space for nothing.
Yes, PCBWay will definitely handle 0201, like any company which has significant production numbers. They use real manufacturing equipment, not only to support modern process nodes, but simply because modern professional equipment pays itself back in production time. And 0201 is pretty standard stuff since early 2000's.
If you have actual miniaturization need, then use 0201 for the cheap CMs, and if you need even smaller, then discuss about it with the CM in advance, not everyone can do 01005. But claims that 0402 is somehow iffy are ridiculous, it's not year 1999 now.
TLDR:
0402 for generic work when size does not matter that much but you don't want to arbitrarily waste space either
0201 for miniaturization but still workable with usual cheap-ish fabs
smaller: work with the CM