The Cup Winners' Cup (CWC, annoyingly now the Club Word Cup is also CWC) ended because it was sacrificed to the altar of money. It didn't make enough money so it had to go, UEFA and big clubs' logic. Let me explain.
As you know before 1998 the CL had only the champions of each country.
The CWC had the winners or runners up (in case the champion had also won the cup) of the national cups.
The UEFA Cup was for teams placed 2nd, 3rd, 4th, etc in the their league (unless they won the cup in which case they would go to the CWC), depending on how many spots each country had.
The problem in the 90s was that football started to bring a lot of money and UEFA and the big clubs wanted to hoard the most of it as possible.
In order to maximise income UEFA had started to shun at the idea of champions of non-rich countries to go far in the competition and at the idea of big clubs not playing each other often.
So they decided to change the format and create the group stages (at first only 2 groups). With only 8 clubs in the group stage finally big clubs could play each other often and everyone was happy, right?
Wrong, what happened (and remember the Bosman law was just implemented and it would take some years to go full force in effect) was that often the champions of big leagues were knocked out before the group stages, like it happened to Barcelona, Stuttgart, Man Utd, Arsenal and Kaiserslautern. This was fair from a sporting merit, those teams were knocked out in the field, but for UEFA this was a disaster - where was all their money?
So they increased the group stage to 4 groups. Now finally all the big leagues had teams there, but now they were also spread out so big clashes often didn't happen until the knockouts. How could this be solved to make even more money?
They knew what they had to do. They had to deturp the competition to allow several entries of the big leagues at once. And they did it. Now the big leagues could put 2 teams in the competition, being a champion was not a requirement anymore - basically making the name wrong - but of course only the big leagues by ranking could have 2 teams in the competition. The smaller, poorer leagues could still only have 1 team there.
Eventually this worked well for money and these leagues started to snowball, getting richer, poorer leagues getting poorer, and bigger leagues got more and more spots for the CL at the expense of smaller leagues that had their champions have multiple qualifying stages to even get to the enlarged group stages. This is where we're at today. It is not a coincidence that since the CL was extended to non-champions only once (Porto 2004) it was won by someone outside the big 4 leagues, and in total only 3 finalists were not from the big 4 (Porto, Monaco and PSG - this last one sort of not representing the french reality at the time with their external financial muscles).
BTW this financial snowballing of the big leagues is also very easily shown in the Europa League: since it was rebranded in 2010 only once (Porto 2011) it was won by a club outside the top 4 leagues. It is more "balanced", as several finalists from other countries existed, but in the end the top 4 leagues' countries are dominating with the wins.
Even in the Conference League, where big leagues generally only have 1 team, was won by teams from those leagues in 2 out of 3 years, with Olympiacos last year since finally breaking the cycle (which is still going in the CL and EL).
So, what has this to due with the CWC? Everything.
Cup competitions are more volatile than leagues, that's why national cups generally have a larger rooster of winning teams than leagues and upsets are way more likely.
So in the CWC sometimes you had odd clubs from an european perspective, mid table teams that managed to win it or reach the final and getting a spot in the CWC.
And it was only 1 club per country, so if a relegation fodder in a big league won it it would be that countries' representative.
This was less than ideal in financial matters and with few big teams in there (deservingly, they should have won their participation by winning in the field) the competition was not generating as much money as UEFA wanted.
In the UEFA Cup this was less problematic because since the beginning it allowed multiple entries per country and since they were 2nd, 3rd, etc in the league it often went to big clubs in those leagues (remember, leagues are generally harder for smaller clubs to get near top than to win a cup competition). So from a financial perspective the UEFA Cup was fine, there were often clashes between money-bringing teams.
The death blow to the CWC was the CL expansion to non-champion teams.
It was already struggling (financially, in sporting merit everything was fine) with "odd" teams, but now the cup winner or runner up often didn't even go here but to the CL as the runner up spot. So it was even more likely that a "big club" that could have brought eyes to the competition skipped it entirely because they qualified to the CL now. See the example in 1998: for the CL the qualifying teams fron the Netherlands were Ajax (Champion) and PSV (2nd place, and it was also Cup Winner). As the cup final was Ajax vs PSV it means that the CWC spot went to a semi-finalist, in this case Heerenveen.
The CWC was thus completely empty of meaning. The Cup Winner did not go there anymore if they did a good league campaign and if the other finalist was the champion / runner up then neither of the finalists went anyway. The CWC literally was scrapping to barrel to find participants.
With even less big teams in the competition and with the supposed participants leeched off to another competition it was from a sporting and financial perspective pretty much dead and UEFA quickly killed it.
The UEFA Cup was affected of course too since it lost the 2nd places and later 3rd, 4th places, etc. But since it had multiple clubs per country there were generally enough big teams to keep it bringing cash. At some point even CL 3rd places in groups started falling in it to make sure at least some big teams kept it alive from a financial perspective (merit alone it not it). It was a failsafe as more and more clubs in the big leagues got spots for european competitions (see how they often get 7 or 8 european spots, almost half a league table) - the big country money has to keep rolling.
TL,DR: The CL expansion made the CWC empty from a sporting and financial perspective. And the CL expansion was made exclusively from a financial perspective. Thus the CWC was killed by UEFA and the big clubs' relentless drive for more money, no matter the sporting consequences.
I liked this competition but in the modern day and age I don't see how it could work with the current model and financial reward of the CL.
Any cup winner or finalist would absolutely ditch the CWC spot as fast as possible if it meant CL money, in the end the competition would be a zombie competition with only flimsy hints of merit (semi-finalists, quart-finalists in cup competitions maybe in some cases) and very little financial backing to it.