I'm looking for a solution myself, so here's some of my thoughts on options:
1) You could go down the complete DIY route, buy a NAS to connect to your home broadband, install a webserver on it and configure it so that external users can connect to it. That way all the files and the web server are in your house in your control.
2) The next option is to pay for hosting and install a photo gallery/content management system and upload your photos there. I had been doing this for quite a while for family stuff, using Dreamhost for the hosting and a system called (imaginatively) "Gallery"
http://gallery.menalto.com/ but it's not been updated in ages and is rather clunky. I could change my gallery system, but it would mean having to migrate the data.
3) Pay for a full solution. Use something like Squarespace with it's user galleries, or Smugmug. Both of which will work nicely with mobile browsers. Smugmug own Flickr, but it's a different solution developed for a different purpose.
4) Go for the full corporate evil and use Google Photos.
The problem is that you can't get something for nothing on the internet now. Google, Amazon and MS own larger chunks of the services available. You end up needing to pay for a service somewhere along the line, but then it's a case of how long till that service is bought up by a bigger corporation. Flickr was independent, then owned by Yahoo, then one of the big US networks, now Smugmug.
I gave up my Flickr Pro account because it wasn't worth paying for for me.
Option 3 is still looking like the least-worst option.
I looked at that Family Album linked above and it's an option, but it is geared for phones. Unless you pay for the premium version when you can upload from a PC. It also looks like they are trying to push their print products too.
The other issue with these services is that they all require every family member to create an account in order to see the photos and sometimes that's a barrier in itself, particularly if you want to share with less technical older folk. "I don't know how to do that, so I won't bother".
I'm not sure that there's an ideal solution.
Personally I want:
To bulk upload from a computer (not a phone),
I don't want to have to install special software.
I want to group photos in albums organised sensibly by year and event
I want to be able to label images
I want to be able to share a direct link to an album, so I can send a link to the latest holiday snaps but I want them to be able to browse the rest of the gallery
I'd like an unfussy user interface with no adverts.
I don't want to have to find another solution in 5 years time.
Not asking much
If anyone has any ideas, I'm interested too.