Is the date alphabetical?
JavaScript (ES6), 101 bytes
Saved 4 bytes thanks to @Shaggy
Returns \$0\$ or \$1\$.
s=&gt;'_414044406550'[[,m,d]=s.split`-`,+m]&lt;(d=`_268328715819832871${6e12-12}`[+d])&amp;d&lt;'_6A9338704'[s[0]]

Try it online!
or Test all dates!
(includes some invalid inputs)
How?
Each month, each day and each year is mapped to an ID in \$[0..10]\$ (\$M\$, \$D\$ and \$Y\$ respectively) according to the following table. We then test if we have \$M&lt;D\$ and \$D&lt;Y\$.
The last part of the lookup string for day IDs is slightly compressed by using ${6e12-12}, which expands to 5999999999988 (19th to 31st).
  M | Month                D | Day                  Y | Year (millennium)
----+----------------    ----+----------------    ----+-------------------
  0 | April                1 | eighth               0 | eight thousand
  0 | August               1 | eleventh           ----+-------------------
  0 | December             1 | eighteenth           3 | four thousand
----+----------------    ----+----------------      3 | five thousand
  1 | February             2 | first              ----+-------------------
----+----------------      2 | fifth                4 | nine thousand
  4 | January              2 | fifteenth          ----+-------------------
  4 | March              ----+----------------      6 | one thousand
  4 | May                  3 | fourth             ----+-------------------
  4 | June                 3 | fourteenth           7 | seven thousand
  4 | July               ----+----------------    ----+-------------------
----+----------------      5 | ninth                8 | six thousand
  5 | October              5 | nineteenth         ----+-------------------
  5 | November           ----+----------------      9 | three thousand
----+----------------      6 | second             ----+-------------------
  6 | September          ----+----------------     10 | two thousand
                           7 | seventh
                           7 | seventeenth
                         ----+----------------
                           8 | third
                           8 | sixth
                           8 | tenth
                           8 | thirteenth
                           8 | sixteenth
                           8 | thirtieth
                           8 | thirty-first
                         ----+----------------
                           9 | twelfth
                           9 | twentieth
                           9 | twenty-*