It's really hard to judge salaries. Here's a salary chart for the MLS in 2018 for example (they publish individual salaries every year) -
Median (ie what half the players earn less than) is $140k, or about E115k. That's not huge money - E2.2k a week. Wouldn't be much of an increase from his time at Rovers I'd say. I don't see what Byrne has done to be in the top half of earners, but in any event you can see it increases very slowly until the top 6/7%
APOEL's budget is around E10m a year, so they could easily match the MLS for salaries. Say two-thirds of that is salaries - that means they're paying E130k a week in salaries. That'd include admin staff, coaches, star players taking a good chunk, etc, etc - but you can see they could probably afford E2-3k a week easily enough.
Obviously covid is going to play havoc with budgets everywhere.
Is the MLS higher profile than Cyprus? I don't know it is. Playing in Europe with a top Cypriot team would have him in some decent grounds (Sevilla, Ajax and Basel last year for example; Dortmund, Spurs and Bayern two years before). No MLS team will get you that kind of exposure.
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