I completely understand you! I have faced all those issues, and it's worst when the designers implement something without asking you first if that's possible to do on web or mobile.
For me, a true web designer is the one that crafts the HTML/CSS, not the one that just puts together some elements on Illustrator/Xd because compared to the actual code that moves the web, that's a child's play and even then... they don't do it right. AND don't get me wrong, I have worked as a graphic designer too using the Creative Cloud suite, so I have been on the other side.
Also for me, hiring someone to make the mock-up and another one to replicate the same but on code is a lost of money, instead of having a front-end developer doing just the web, or both! since software like Adobe Xd is extremely easy to use and building a mock-up with it can be fast.