It sounds to me like you're looking for something that will cost $$$ and potentially take at least as much time maintaining as you currently spend. I did look around for membership software solutions, driven mostly by email management, but ended up deciding that the cost was ridiculous.
We also use a simple Google Sheet for our roster. Because all of our memberships renew at the same time, we do not have much to maintain. Invoices go out once a year, in late November or early December, which our Treasurer and I use as an excuse to get together with a few beers and pizza. Sort the spreadsheet by Member Type (Single, Couple, Spouse), then bulk copy and paste email addresses from the sheet into a PayPal invoice 20 or so at a time. Singles and Couples get invoiced for different amounts, and Spouses don't get invoiced at all. This literally takes us less than an hour once a year. Then we check back in about February, see who has paid with PayPal (or cash/check directly to the Treasurer), and update the Year Paid column on the spreadsheet. A couple more beers and another pizza. If there are stragglers who haven't paid, it is a simple matter to sort the Sheet by Year Paid and send a reminder email by copying and pasting email addresses in bulk.
New members require a little bit of manual work: adding them to the roster spreadsheet, adding them to the Google Group, and if they request it, approving their access to the Members Only page of our web site and our FaceBook Group.
The biggest hassle we had was managing our email distribution list. We were using various Comcast email accounts with forwarding filters set up. That got way out of hand maintaining multiple filters to add/edit/delete people's addresses as they came and went.
The solution was the Google Group. Set it and forget it. Enter addresses once as people join. Then whoever you want to set with the authority to send club-wide emails, has only to send one email to the Group address and it goes out to everyone. No one gets accidentally left off.
We have membership card capability by creating a PDF with two editable fields (Name, and Year). It is uploaded to the Members Only page of our web site which requires a login to access. (It is a Google Site linked to our domain.) I honestly don't know if anyone actually uses it, the only purpose of which would be to get a discount at one of several area homebrew shops. But they all honor our club discount without the need for a card.