How to remove Golf in Garmin Connect?

This isn't a "problem," but rather an option/feature that you simply do not make use of. Others may never go swimming with their watch or utilize the equipment tracking features. Currently, there is no option to remove features that you do not use from Connect Web and Mobile versions, and I would not expect that to change as it would defeat the purpose of these menus.

on the mobile app it's even worse, there's a second menu that allows you to download golf courses. Useless for most people.

All other activities are submenus of the "Activities" menu (it's coherent).

It gives the impression that the applications were poorly designed ... it's a shame (especially if we read the different topic of the forum, the problem has lasted for several years).

In garmin connect web version there is also something inconsistent in the management of the profile.

I don't see any golf on either Garmin Connect for Android nor the web version. I wonder if it's either to do with what kind of watch you have (I use a VivoSport) or in your activity settings?

You can change these under "settings > public profile" but I don't know if it will make any difference. Do you have golf enabled there?

I just completed my messages by photos.

My watch is Garmin Fenix 5 Sapphire Edition. I never installed Golf app. The Watch is up to date.

In Public Profile i don't have Golf Option checked :

And in Garmin (Android), the golf app is in the deleted apps category.

Golf is sort of it's own unique thing in the Garmin universe. It has a functionality, options, and information that is different from all other tracked activities, so it is treated differently. Probably good news for you is that Garmin appears to be making a Golf application, so perhaps all Golf information will be removed entirely from Connect Mobile in the future.

There is a rather prominent golf section in Garmin connect. Two sections Download golf courses and Garmin Golf with their own colour theme.

I do not play golf and would like to remove this section.  Ideally if I could replace it with quick access to activities I do engage in, that would be great.  If this is not possible, then simply removing the golf will be fine.

Surely this is a setting I can apply somewhere?

As other users stated, this probably depends on a watch you have. Mine doesn't show golf, so I can't test it, but you can use one of AdBlock web-browser addons with rule like this:

connect.garmin.com#?#LI:-abp-contains(Golf)

This blocks LI element from menu on the left, if there is more golf somewhere you need to block others elements similar way.