I added SSL to a website, but blog-hosted images stay as non-ssl and won't show

Howdy!

I’m experimenting with SSL, trying to stop being a newbie with this all. Successfully installed, got it working and all. Got it working on several domains, with either wosign (that is giving free certificates) and letsencrypt.

However, I noticed something troubling, could I ask you, please, if:
– this is normal to have it happen
– there is a solution to this problem other than a mass database search-and-replace

Here’s my problem.
The blog is working perfectly without SSL, as http://
The posts contain image attachments, added through the wordpress library, that follow a simple structure, such as:
<img class="aligncenter size-full wp-image-11209" src="http:// (blogurl.net) /wp-content/uploads/2016/04/estimedesoi.jpg" alt="estimedesoi" width="300" height="366" />

But images won’t show when I use the https:// version of the blog, because mixed contents are blocked, only the secure contents are loaded, say my browsers (tested with Firefox and Chrome, only good old IE loads everything without second thoughts).

At this point, I know I could run a simple operation in phpmyadmin to replace all http://blogurl.net/wp-content/ with https://blogurl.net/wp-content/ in wp_posts.
However, that also means that the day SSL is turned off, *poof*, the images in posts are broken again.

Before I do that search-and-replace in the database, I wanted to ask on the forums first.

I mean, doesn’t it look weird, that wordpress won’t detect it runs from a secure session and adapt its own posts output ?
It strikes me as odd and illogical that switching a blog to an SSL version, an operation praised nowadays, would result in the obligation to update all the old posts about their media embeds.
And isn’t it weird that the browsers consider this mixed content while it’s served by the same website anyway ?

Thanks if you, the more competent people, have an idea on this 🙂