Player recruitment thoughts:
When it comes to CL/No CL etc, player recruitment has plenty of nuances.
For example, Kylian Mbappe and Pierre-Emerick Aubameyang have both mentioned their admiration for Real Madrid. It's unlikely that Real would sign both. If they both want to go, they will be insanely expensive. However, if you throw 70+ million at Dortmund/Monaco and offer 250K+ a week, you could get them. In those sort of scenarios, you would hardly have any competition either. I mean, there are dozens of teams in the CL, but how many would offer that much money? Man United paid 90 million for Pogba. Who was their competition? If no one bids the amount you do, and he's interested, well he's yours. If City/United don't get CL but throw 85 million at Monaco for Mbappe and offer him 250K, he's going there. Likewise, if Griezmann is thinking of leaving, and City/United throws 85 million at Atletico and offers Antoine 300K a week, one season out of the CL doesn't matter. He may love to play at Bayern, Barca, or even Arsenal/Liverpool/Spurs/Chelsea but if those clubs don't offer the money, he's not going there. CL is basically irrelevant at that point.
Where CL really matters is for aspiring stars where no one is throwing around obscene money. For example, we offered Mkhitarian good wages (maybe more than Dortmund), but the CL and Klopp's run there probably played a huge factor. Of course, if Shakhtar had set a higher price (let's say 50 million), and we paid that and offered him 200K a week, he's ours. But if us and Dortmund both bid 25 million (accepted by Shakhtar), we're going to have to pay some damn fine wages without CL (and at that point, we had finished 7th, 6th, 8th, 7th the previous 4 seasons). You can see why he would pick Dortmund.
When it comes to Arsenal, whoever their manager is, and their situation is, if they throw around silly money, players will come, CL or not. However, where they could have a problem is if they're looking at players like Naby Keita, Jonathan Tah, Thomas Lemar, etc. If CL teams come in for those players, Arsenal will have a tough time getting those players at reasonable prices (by modern football standards) if CL clubs come in. They'd have to blow CL teams out of the water, which makes team-building really difficult (constantly overpaying). Consistently overpaying is also not what Arsenal have done over the years.