Any invisible jet is more or less going to be some type of stealth technology. I think that's pretty much a given at this point. And given the timeline between WW and BvS/JL that's more than enough time for Diana to have gotten some type of pilots licence to explain how she knows how to fly a plane. But if she flies, and I'm still not 100% sure that's been defined yet, then the jet is useless.
I don't think it's useless. In the comics, there were many Avengers who could fly (Iron Man, Thor, Vision, Ms Marvel etc) yet they all travelled around in the Quinjet. Not sure how big they are in the movies to accommodate so many passengers, but in Agents of SHIELD they have the Zephyr One which can cloak and decloak, rendering it invisible.
WW could fly, but what if she has to rescue a mass of people? She wouldn't be able to hold them all and she's not fast enough like the Flash to go in and pull them out all one by one. A jet would come in useful for rescue missions. It would also come in useful for transporting around other members of the Justice League. Yes, Superman could fly too, but what if they actually want to be going over strategies and plans? With WW or Superman flying off somewhere else, they can't exactly sit in on the discussions. It wouldn't be the same just wearing an earpiece either, especially if they have to look at visual information.
And if the League are going on a secret mission, then they don't want to be flying in detected, which both WW and Superman would be if they're not cloaked by an invisible shield. They might all want to get into position covertly so they can retain the element of surprise and appear at the appropriate time.
The jet suffers from the same stigma as Aquaman had from the SF show, only combined with people's perceptions from the Carter show. Which... I don't get. Do people really think that in today's world of film the jet would look like a pencil outline or a toy with a doll in it?
This. I think this is similar to before Batman 89 came out. People probably still were influenced by the 60s Batman show and also thought many things could be hokey. People don't think that now about Batman. With WW, all it would take would be one movie to change people's perceptions of the invisible jet. As mentioned, it's been done in many other films or TV shows already such as Star Trek (with the Romulan bird of prey) or Agents of SHIELD with the Zephyr One. None of these seem ridiculous at all and no-one even thinks of the invisible plane from Lynda Carter's TV show.
The only issue is where she gets it from. The obvious answer is Bruce but I feel him being the one responsible for everyone's tech is gonna get old pretty quick.
I don't think she needs to get it from Bruce, although he would be the obvious one. But if WW is on a mission to steal a stealth plane from an enemy (they could be anyone - not necessarily even a superpowered enemy but someone like the Russians) then if she already can pilot a plane, she could take it at the end of the movie.
Then, while being chased by enemy combatants, she could be playing around with the controls and find the cloaking button which would turn the plane invisible. But what could happen is that, while evading some of the pursuing planes, she might discover that one of them is still able to detect her because that too has the ability to either track a heat signature or can bypass the cloaking technology. So maybe the plane fires upon her and damages the cloak so that she discovers, when she finally gets away and lands, that the plane is no longer able to decloak but is stuck in cloaked mode. This would give her the invisible plane in a plausible way that audiences could buy into, because they've seen the process by which it happened. It wouldn't seem that she just had an invisible plane out of nowhere.
