Only rarely did I not enjoy a story in those Asimov Best SF volumes,but I wouldn't remember that story in particular

The Girls From Planet 5-Richard Wilson (1955)
Second Printing 1967 No cover credit

America elected its first female President in 1988 and by 1999 the women run the country.All except Texas where men still rule the old fashioned way.Suddenly space ships appear across America.Out emerges gorgeous female Amazons claiming a friendly visit but evading most questions.Texas reporter David Hull is suspicious and tries to get some answers.The US government immediately calls them our friends even after the accidental incineration of West Alexandria when the first star craft landed

Richard Wilson was a tongue-in-cheek writer popular in the 50s.So what was I going to read here-An old fashioned goofy chauvinistic tale or a novel that was actually progressive for its time? It was the first of the two
A female dominated government is depicted as extremely pacifistic and indecisive.As the male dominated Texas governor states-Females are good homemakers and keep people confortable.But if a burglar enters the house they'll barricade themselves in the bedroom,hide under the covers and wish they had a man to protect them.Texans call the rest of the US "Biddyland".The Amazons,called The Lyru, arrive,apoligize for destroying a Washington DC suburb and are welcomed with open arms.All except Texas who mobilize their Rangers and will attack any Lyru who enter their state.Reporter David Hull is able to convince one Lyru to defect and shows her earthly customs while trying to learn about Lyru's society and the real reasons they arrived.
This 1955 tale actually got a few things right.It mentions the Russian Federation of Liberated States and The European Economic Community.Also,in Texas,horses are the prefered and manly way to travel but horses' gear have built in ridie-talkies and mini-bars.Otherwise there is no other mentions of world events or scientific progress.It does say after 2 moon launches that failed,the female government shut down NASA as a waste of money.
Its a novel thats meant to be light-hearted and ridiculous.Cowboys vs Amazons.Men make bad house-engineers and women don't belong in the Pentagon.

I've mentined liking the couple of Richard Wilson novels that I've read, And Then the Town Took Off (half of an Ace Double; not sure it's ever come out as a stand-alone) & Thirty-Day Wonder, which in turn inspired me to look for Girls from Planet 5 for several years. Once I finally secured a copy, though, the first few pages failed to grab me.

Your description is exactly what I expected; I need to dig the book up & try again, obviously.

As for the cover, looks like Kelly Freas to me.

Was that the original novella or the "novel" that was made up of that novella and its two sequels?

The Best Of John Sladek-1981 1st Printing
Artist Unknown

A short story collection from tales previously appearing in Keep The Giraffe Burning and The Steam-Driven Boy

Contents:
The Secret of the Old Custard (1966)
The Poets of Millgrove, Iowa (1966)
The Best-Seller (1966)
Is There Death on Other Planets? (1966)
The Happy Breed (1967)
The Singular Visitor from Not-Yet (1968)
The Short, Happy Wife of Mansard Eliot (1971)
The Momster (1969)
1937 A.D.! (1967)
The Transcendental Sandwich (1973)
The Steam-Driven Boy (1972)
Elephant with Wooden Leg (1975)
The Locked Room (1972)
The Face (1974)
The Bright Side (1975)
Getting There Is (n-1/n)th the Fun (1975)
Mr. and Ms. America (1975)
The Paradise Problem (1975)
Empty Promise (1975)
What Changed Doyster's Mind (1975)
Assessment (1975)
Handout (1975)
Pax Gurney (1975)
Art News (1975)
A Fable (1975)
Utopia: A Financial Report (1975)
Luck (1975)
Utopiary (1975)
A Picnic (1975)
Space Shoes of the Gods (1974)
The Purloined Butter...*dg*r *ll*n P** (1972)
Pemberlеy's Start-Afresh Calliope or, The New Proteus...H. G. W*lls (1971)
Ralph 4F...H*g* G*rnsb*ck (Hugogre N. Backs) (1973)
Engineer to the Gods...R*b*rt H**nl**n (Hitler I. E. Bonner) (1972)
Broot Force...*s**c *s*m*v (Iclick As-I-Move) (1972)
Joy Ride...R*y Br*db*ry (Barry DuBray) (1972)
The Moon Is Sixpence...*rth*r C. Cl*rk* (Carl Truhacker) (1981)
Solar Shoe-Salesman...Ph*l*p K. D*ck (Chipdip K. Kill) (1973)
One Damned Thing after Another...C*rdw**n*r Sm*th (A Co-ordainer's Myth) (1973)
The Sublimation World...J. G. B*ll*rd (J. G. B-) (1968)

John Sladek was born in the U.S. but spent much of his life in England.In was part of the 1960s New Wave movement.His novels included The Reproductive System (AKA Mechasm),The Mueller-Fokker Effect,Roderick,Roderick at Random and Tik-Tok

As you can see,the stories are very short,6 or 7 pages on average.Known mostly as a satirist.
The last third of the book contains parodys of other SF writers.These I really enjoyed.The Gernsback,Heinlein,Asimov and Clarketakeoff are hysterical.The other parodys are fine

I really struggled with his stories from the 60s to early 70s.They had some fun parts and good ideas but were mostly nonsense stories.The Steam Driven Boy was were I began to like his work. The story is about a despotic President who wants Americans to be happy with less and less.For instance his Horsemeat commercials-"Ralphie loves it,every bite.Why don't you try horse tonight?"

I'd say I really enjoyed half the stories,did not like the other half.There was no in-between

I've read a couple Zane Grey novels... they're pretty good, but not all that deep, certainly.