QGIS point sample tool faster than in PostGIS - how to improve query?

Question

My points table has 246,338 features and the polygon table has 43,957 features. I am relating the values of one attribute from the polygons to update a column in the points table.

This question is similar to Speed up point sampling with ST_Value function in PostGIS, raster/vector overlay but that question specifically addresses point sampling of raster, not polygons, using ST_Value, and the optimum raster tile storage size.

Both tables have had maintenance performed and have spatial indexes that are updated. The polygons are records of fertilizer application rates applied to a field and are fairly messy with plenty of overlaps:

The polygons table has been checked for invalid geometries and those have been fixed. Running ST_NPoints tells me there are an average of only 5 vertices per polygon so ST_Simplify is not necessary. The average polygon size is 112 m2 and range from 0 (only 60 of them) to 826 m2. I cannot dissolve (ST_Union) polygons on anything because I need the unique numeric fertilizer values from those individual polygons.

I've tried two queries in PostGIS, but have abandoned them after running them for a few hours:

Query 1:

update points
set fert_rate = rate
from points a, polys b
where st_intersects(a.geom, b.geom);

Query 2:

with pnt_sample as (
  select rate
  from points a, polys b
  where st_intersects(a.geom, b.geom)
)

update points
set fert_rate = pnt_sample.rate
from pnt_sample;

Having QGIS read the tables from the DB and run the 'point sampling' tool takes only a couple of minutes. However this is not the optimal solution for many reasons, particularly because the output must be saved as a shapefile, headers are truncated to 10 characters, and then must be imported back into the DB, header names fixed, etc.

My questions are:

1. What is QGIS doing in its algorithm that I don't have in my query?
2. How can I incorporate that advantage into my approach so I don't have to go outside of my database?

I would like to keep any suggestions/solutions within PostGIS/QGIS.

I'm using PostGIS 2.4.1 and QGIS 2.18.14 on a Windows 10 machine.

EDIT: In response to John Powell and Thingamubob's comments, I ran EXPLAIN ANALYZE on the two queries above. Both tables had vacuum/analyze performed just prior to running EXPLAIN ANALYZE. Here is the output for the result of the first query:

And here is the output from the second query, the one using the CTE:

Each of these EXPLAIN ANALYZE queries took about an hour to run.

Answer 1

Does it work any better with an inner join rather than a full outer join? I expect an inner join would make better use of the indexes, where the full outer join would assemble every possible combination and then evaluate the results.

(This was edited with a presumption that you have a key column called "id" with a single-field primary key or a unique index)

with pnt_sample as (
  select a.id, b.rate
  from points a
  JOIN polys b
  ON st_intersects(a.geom, b.geom)
)

update points p
set p.fert_rate = ps.rate
from pnt_sample ps
where p.id = ps.id;

Alternatively:

update points
set a.fert_rate = b.rate
from points a JOIN polys b
ON st_intersects(a.geom, b.geom);

Related: Acquiring ArcGIS-like speed in Postgis and https://stackoverflow.com/questions/17946221/sql-join-and-different-types-of-joins

If polygons overlap, then choosing the correct value may require some grouping.