I would recommend adding the pickup address in the confirmation email that is send to the users.
You can find this by going to Settings > Notifications in your Admin and find the "Order confirmation" notification.
Seeing the raw code that builds up this email notification can be very daunting, so an app that lets you visually edit the emails can help. I can recommend OrderlyEmails.
I'm not sure there is a really nice way of not displaying the actual address, there is however the following slightly cumbersome option:
You can create a new location you can use for pickups at Settings > Locations. You're allowed to create a location with just a country name (and state if you're located in US). You can even fill in City or zipcode data if you feel comfortable enough to provide that data. When creating you'll get a warning message saying the address information can't be verified and that you might have issues with printing shipping labels, but that's not an issue for a pickup location. You also can't mark this pickup location as your "default" location when there is no address info.
Second, you'l have to transfer all your inventory to this location. If you go to Products > Inventory you can view products and their inventory on a specific location. This filter for location is located at the top right of the list of products.
If you select all product (top checkbox will allow you to select all), under "More actions" you can find the option to "Create transfer". You'll have to walk through all your products and transfer the available amount to the pickup location (the cumbersome part).
If you do this, and there is no product stock anymore at your home address location, your home address won't show up anymore as location for the ordered products.
I hope this helps.