And let's get the most obvious one out of the way. GOG should not be emulating any of Steam's single-player DRM at all.

NONE.

OK, maybe one: better Linux support.

Are you saying that GOG is not successful?
;-)

Better Linux support.
Actually being able to find, edit, and make use of formatting in reviews.
Ability to kick demos and other non-content out with little managerial oversight.
More than two ways to view a game library.

An actual use for profiles and friends.
A better front page.
The return of curation type features to the users.
A better overview for reviews instead of having to scroll to the bottom of the page like a complete plebian.

Automatic invoice and billing of partner companies.
Removing support for quaint obsolete systems.
A proper game tagging system.
A command line interface, for days when I don't feel like looking at GUIs.

Balatro.
Opt-In experiments/labs, instead of surprise A/B tests.
Proper community hubs for the games, which the gamecards can organize around.
No seriously, a way to ignore game suggestions in the store.

Better Linux support.

Galaxy is not compatible with Linux, but I don‘t care. I don’t want it anyway.

On the other hand I can see many games that are available for Linux on Steam, but can only be found for Windows on GOG. I end up skipping many games I would have bought without a second thought otherwise

Get rid of GOG Galaxy.
Steam doesn't use GOG Galaxy, and it's the one thing about Steam that I can get behind.

Isn't that because Steam uses Steam? GOG Galaxy is a competitor.

I would like to be able to mark games owned elsewhere like I can do on Steam.

I don't think that would do much to make GOG successful though, but a nice feature for users with many games in many stores.

Better yet, a list of aspects of steam gog shouldn't emulate:

Achievements
(sorry not sorry)
The current ones are little more than pavlovian dopamine drips made to hook people into buying more games.

Cloud Saves
Case in point: gog's more recent stickies about cloud save limits
Saves are best stored locally or on (if any) free/non gated game save backup services.

Regional Pricing
Some of us have to pay 15 bucks (on sale) for a recently released game while some others pay 1/3 or less of that price.

Better search, wishlist, client, larger achievement support for games.

Although, differences are also a thing that appeal to me and raise the interest.
So do your own thing and do it well!

+1 native linux games.

Better Linux Support would would be great.

Having the games up to date, like seriously, GOG shouldn't allow that shit unless there a good reason for the delay

Now, that alone wouldn't really push GOG forward that much, but it would be a first step. They should push to have the GOG logo in trailers, tons of games only get advertised as coming to PS, Xbox, Steam and Epic even if they're ALSO coming to GOG

Which aspects of Steam should GOG be emulating to be successful?

Maybe the better question(s) would be:

Which aspects of Steam did/does GOG try to emulate so far?

And

How detrimental has that proven to be for their success?

Almost nothing, just try to bring more games to the catalog, that's it.
I just wish GOG and the publishers would be more careful with packages of old games like it was years ago (better compatibility with modern systems, all documentation included (manuals, ref cards etc), all expansions included...), something that steam never cared and never will care. When we talk about classic games, there are more broken ones there than here.