Do Warner Bros. and Disney still dominate the TV toon market like they did in the late 20th century?

This may or may not be relevant. I'll drop it here and let other's decide.

The director of the Disney documentary "Waking Sleeping Beauty" is working on a new documentary, based on the life and career of lyricist Howard Ashman (The Little Mermaid, Beauty & The Beast, Aladdin).

It will be on Disney Plus next month.

I would like for Cartoon Network or some other cable/satellite network to air a documentary about Walter Lantz.

NBCUniversal would have to produce it as they own Lantz's creations and his old studio.

They recently launched their new streaming service Peacock, which has a free option with ads. If NBC were to produce such a documentary, they would likely release it on Peacock.

Peacock launched with a couple exclusive animated shows, such as a Where's Waldo? series and a series about Cleopatra in the future.

That's true. However back in the late '90s Cartoon Network did air old Woody Woodpecker cartoons.

Probably because Cartoon Network had a license to air Woody Woodpecker cartoons back then, but not now.

Networks in general prefer to produce their own content, rather than pay another company for shows.

From what I can find, Cartoon Network mainly features their own shows or other WB animations. They only have 2 acquired shows: Lego's Ninjago and the French series Power Players.
https://en.wikipedia.org/wiki/List_of_programs_broadcast_by_Cartoon_Network

You can't expect their programming, or anything honestly, to stay the same for 20 years.

Yes, exactly. I forgot the name of the programming block CN had in which they showcased classic Woody cartoons but they were definitely on there.

Do you know which series of classic cartoon shorts they should bring back and air on CN or some other cable/satellite TV network for historical purposes?

I can think of a few:

Beany and Cecil
Heckle and Jeckle
Mighty Mouse

Most of my favorite older cartoons are already available on HBO MAX, Boomerang, or Disney Plus, with the exception of Tex Avery's "The ___ of Tomorrow" shorts.

ViacomCBS does own the Terrytoon library. It is possible that those cartoons will be available to watch on CBS All Access when it relaunches later this year.

Probably it was Toonheads.
http://en.wikipedia.org/wiki/ToonHeads

Yes, it was either Toon Heads or The Tex Avery Show.

Even if the networks did bring back classic cartoons, I doubt they will be popular as modern cartoons.

For example: HBO MAX and Disney Plus, as well as Amazon Prime and Netflix, all have classic films (1920s-1970s) available on their services, but they are not widely viewed as modern films and shows. There are numerous reasons for this: different social views, black & white, not aged well, etc.

This applies to the cartoons as well. HBO MAX and Disney Plus have many cartoons from the 30s and 40s, but they are not watched by the general audiences. I watched them, but they are not exactly memorable and not really funny anymore (sometimes amusing, though).

At best, people would watch the animated shows from the 80s-2000s for nostalgic reasons and nothing else beyond those years.

A special mention here:

As the San Diego Comic Con 2020 has been cancelled this year due to the pandemic, Comic Con International has announced a online version, Comic-Con@Home, that will be free to everyone on YouTube July 22-26

"Once the daily schedules go live, you’ll be able to read the descriptions and decide what you’d like to see. The majority of our panels will also be available AFTER the July 22-26 dates, although there are some that may have a limited time period attached to them. Each panel will contain a link; when you click on that link, you’ll launch another page that has the panel video embedded on it … that video will be available on the date and time listed. When it’s available, simply push play and watch! It’s that simple."
https://www.comic-con.org/cci/2020/athome

All the panels will also be available on the Comic-Con YouTube channel, https://www.youtube.com/user/ComicCon

HBO MAX will have an Adult Animation panel on Fri, 7/24 3pm PST, while Disney+ has their Marvel's 616 documentary panel on Thur, 7/23 1pm PST and The Phineas & Ferb The Movie panel Sat, 7/25 12pm PST

Meanwhile...

DC will hold its own 24 hour fan event, DC Fandome, on Saturday August 22.

"Accessible for 24 hours at DCFanDome.com, the global event will immerse fans into the DC Multiverse, with new announcements from WB Games, Film and TV, and comics, as well as an unprecedented opportunity to hear from the casts and creators behind your favorite feature films and TV series (current and upcoming), including: Aquaman, The Batman 2021, Batwoman, Black Adam, Black Lightning, DC Super Hero Girls, DC’s Legends of Tomorrow, DC’s Stargirl, Doom Patrol, The Flash, Harley Quinn, the SnyderCut of 2017's Justice League, Lucifer, Pennyworth, SHAZAM!, The Suicide Squad, Supergirl, Superman & Lois, Teen Titans GO!, Titans, Watchmen, Young Justice: Outsiders and, coming this fall to theaters worldwide, Wonder Woman 1984.

https://www.dccomics.com/blog/2020/06/16/welcome-to-the-dc-fandome

What company currently owns the rights to the old Harveytoons cartoons?

A quick Google search says NBCUniversal's DreamWorks owns the Harveytoons
https://en.wikipedia.org/wiki/Harvey_Films

None of the Harveytoons are available on NBCUniversal's Peacock service.

They do have an ongoing/recent agreement Netflix with the rights to Sabrina and the Harvey Street Kids and their recent excellent/well-made series on Netflix.

Not sure what other Harveytoons related characters are in the project phase in the Netflix pipeline.

HBO MAX announced new details for the next Adventure Time special, Distant Lands: Obsidian, which will be about Princess Bubblegum and Marceline going on a sci-fi adventure while they figure out their new romantic relationship.

https://www.polygon.com/animation-cartoons/2020/7/24/21337454/sdcc-2020-adventure-times-obsidian-panel-bubbline-marceline-bubblegum-details-plot-characters

When I was a kid the Harvey Comics were a strange exotic world. My dad got me some from a railway station or something. I’d never seen anything like them, our comics were nothing like that, for a start they weren’t in colour.

We never saw the cartoons. I remember reading the adverts for Caspar cartoons in the comics and feeling quite envious. This was the seventies we had two channels and a black and white telly.