How do I change the default activity type on a custom activity?

You can't change the type in the activity, only after it was recorded. But since you want it to be Cardio you can copy a Cardio activity (you have to redo everything, but at least it's possible, unlike for example for the type "Inline Skating"). There are a lot of types that can't be copied and are unavailable directly.

You can send a request that we would like to be able to chose any available activity type for our custom activities:
www.garmin.com/.../

One possibility is hacking the FIT file of the activity profile, that is stored in the folder //GARMIN/Sports. However, you'd need some tools like the FIT SDK, or the FIT File Repair Tool, and a good knowledge of the FIT file format. A kind fellow forum user like for example Former Member or seilogramp could perhaps help you with that.

Another possibility is changing the activity type subsequently in Garmin Connect, always after saving and syncing it.

There is also the Connect IQ app Any Activity | Garmin Connect IQ that allows assigning sports not directly supported on your watch, but since I never used the app, I am not sure whether it would help in your case with the already created copy of a strength activity. 

If I do decide to delete the current activity and create a new one, will my stored activities still remain in Connect and then I just have to manually re-categorize them?

I only have 3 weeks worth of data so not the end of the world if I lose them but it might be better just to do that instead of going through manual workarounds or hacks to the file itself. 

trux said:

acking the FIT file of the activity profile

Interesting, so I could make myself a native Inline skating activity.

Done and synced activities remain as they are even if you delete the activity profiles you have setup on your watch. They are just templates about how to record your activities and what data to store with them. Most of them have some unique aspects to their specific activity types.

KrapfenKringel said:

Interesting, so I could make myself a native Inline skating activity.

If you want to hack it yourself using the SDK, I posted some hints a few years ago in the thread Backcountry Skiing Activity Profile - Instinct - Wearables - Garmin Forums. However, if you do not have any experience with coding, you should better stay away from it, to save you some headaches. Using the FIT File Repair Tool may be easier, though you'd probably need the paid version.

Is it possible to activate step tracking in an activity that has this disabled this way?

Surely yes, but you'd need to find out the field controlling it. Most likely the SDK will show it as one of the "unknown" undocumented fields.