The NFLPA has been trying VERY hard to get that franchise tag removed, but the owners will never go back on that one. That's one of those things the owners just will not move on. It's what really keeps the market from getting just...ridiculously out of control. How often does a truly elite player come on the FA market?
Kirk Cousins was hardly truly elite and got a 3 year fully guaranteed(for skill AND injury) deal. And then he's continued to get them.
So imagine no franchise tag? Bosa is hitting the FA market, Jones...and instead of Nick Bosa looking for a 35M deal, he could very possibly get a 40M AAV deal. But instead he's negotiating against deals signed by the other top edge rushers(his brother, TJ Watt) who were only negotiating with one team.
Watson, a guy with a terrible past and about as unattractive a player as you could ask for wasn't even on the open market and still blew the market up with a 5/230 full GTD(injury and skill) contract that forced other teams to not quite match it, but come close.
If these players were FAs after their 4 or 5 years, the salaries of the top players would balloon that much more and for obvious reasons the owners don't want that.
Getting rid of the franchise tag was a non-starter in 2010-11, it was a non-starter in 2020-21 and the next CBA which runs through 2030, it'll likely be a non-starter by then as well. It'll take the players willing to sit out a lot longer than they ever have to get significant changes.
I think your solution makes some sense, but I still don't think the owners will ever agree to it. Adams would have been able to leave without the Packers getting a 1st and a 2nd for him.
A compromise is probably the best you'll be able to do. I think at most you should be able to tag a player once, it shouldn't be an exclusive rights and you should probably lower the value of the compensation. 2 1sts is just more than most teams are willing to give up for most positions.
The franchise tag is certainly abused, but I don't think it's going anywhere.