in my opinion, and in order with expenses...
first try some new collets.. ... mightbe that solve it already...
next compare the regrinding price with a new spindle,,, I don't know the regrinding prices, but a new spindle is just new,,
also you can consider a small water cooled spindle, you can get for about 250...
65mm 0 8KW ER11 Four Bearing Air Cooled Motor Spindle and Inverter Drive VFD CNC | eBay
endmills under 2 mm dia... need high rpm... under 1 mm sure the 5-6K rpm your machine does right now, just way too slow..
datron using 60K rpm spindles for metal.. so the 5-6 K rpm sounds really slow..