Actual Assembly Yields -- 0402 vs. 0201 / 01005
Our passive supplier ... told us ... to switch to 0201s. They are going to phase out 0402s

The real problem here is the supplier, not 0402s going away.  There are plenty of companies that make, and will continue to make, 0402 and larger resistor packages.  There are also plenty of assembly houses that can handle, and will continue to handle, 0402 and larger.  Resistors in 0805 package are still quite common.  You've apparently found one, but I haven't run into a vendor nor assembly house that can't or won't handle them.

Switching technologies because this one supplier is dropping something is the wrong reason.  Switch if you need the small size for your purposes.  Otherwise find a real supplier.

As for the reliability, that depends on the assembly house.  There is no reason 0402 and 0201 resistors can't work perfectly well if the assembly house is competent.  Pretty much anyone can slap 0805 parts on a board.  Starting at 0402, the process needs to be better controlled to avoid failures, but that is certainly doable.  You should give the assembly house a tester, and they only deliver you units that pass the test.  Failures are on them.

At 0402 and below, you also have to think about temperature balancing between the two pads of the part to avoid tombstoning.  Reliability isn't all just someone else's problem.

Another issue is the power dissipation.  With 0805 parts, you can largely dismiss power dissipation limits for ordinary signal circuits.  For example, 5 V across a 1 k&Omega; resistor causes 25 mW.  That's nothing for a 0805 package, which can usually handle 125 mW.  It starts getting significant for tiny packages.  When using really small parts, you have to think about power dissipation in cases you might be used to just dismissing it out of hand.

However, none of this should keep you from using 0201 when there is good reason for them.  One supplier of the many many out there dropping 0402s is not a good reason.