Excellent question and good to hear people positions on this. I have been recently debated a similar (*) dilemma, having upgraded my turntable, which now has upset the balance in my system - as I agree with the proponents of the balanced approach.
But, as
@Fandango Andy mentioned, some components do punch above their weight, and prices fluctuate so that just using RRP is hardly an exact science...
In my case, I now find myself with a near £3k turntable matched to amp and speakers which are roughly around £1k each.
My initial thought was to upgrade the amp (cue overly technical discussions on power ratings at various impedance levels to determine which amp has the better grip on speakers) - but also I believe the speaker first theory has some weight. If your turntable set up is able to dig out enough information, if the phono stage is capable of transmitting that with enough fidelity, possibly a combination of a "sufficiently capable but not necessarily super duper" amp and higher end speakers capable of rendering that information in finer detail would be preferable to a better amp/average speaker combo?
Going back to the component punching above their weight, synergy etc - in my case I never really felt the need to upgrade my £1k Focal Aria - which were actually £700 when I bought them only two years ago - as I think they work great in my space. Although I'd love to see what a more capable speaker could do, I can't see any others in my budget range that would fit the bill in terms of sound signature (detailed but definitely not bright and with "boogie factor" as I do listen to a lot of rock, funk, etc) and positioning (awkward room, so ideally need front-ported) to justify a serious money outlay. In a "money is not an object" scenario, I'd have the Proac D2R tomorrow, but in the real world, they'd be a serious stretch , as they are £3.6k new...
* obviously that was a work in progress comment which I also obviously accidentally published 😆