This is more a few comments than an answer (since the question seems well answered). I assume that the pair $(a,b)=(1,1)$ was discarded, it would give a value $\frac{\sqrt{2}}{2}$ outside the range of the rest.

Taking instead the $10^6$ points in a quarter disk of radius 1128 gives almost the same bin sizes (maybe not surprising since there is a good overlap). This includes the features that the bin 0-0.005 is smaller than the rest and that 0.330-0.335 is rather deficient and then 0.335-0.34 is higher then average.

This is an indication of a slight repulsion from simple fractions. I repeated the experiment using 2520 bins and using rounding. I also used $0 \le a,b \le 5000$ giving about the same expected number of points per bin: 6030 or in my case 3015 since I only used $a<b$ (the situation being symmetric.)

The least filled bins were $\tiny{[0,0],[1/3,2187],[1/4,2479],[1/6,2761],[2/5,2773],[1/5,2774],[275/1008,2865],[229/1008,2875],[323/840,2891],[229/1260,2893],[1/8,2895]}$ $\tiny{[3/8,2897],[1/7,2897],[3/7,2900],[2/7,2902],[97/840,2906],[155/1008,2910],[1/10,2913],[3/10,2915],[37/630,2925],[59/560,2927],[139/315,2928]}$ $\tiny{[221/560,2936],[127/720,2939],[1/9,2939],[4/9,2940],[199/630,2940],[2/9,2940],[877/5040,2947],[611/1680,2948],[1/315,2948],[157/315,2951]}$ $\tiny{[31/360,2952],[229/2520,2956],[97/1260,2956],[5/14,2959],[1643/5040,2960],[3/14,2960],[1/14,2962]}$

Here 275/1008 is nearly 3/11 and 229/1008 nearly 5/22.

The most filled bins were

$\tiny{[277/720, 3123], [83/720, 3124], [229/840, 3139], [1259/2520, 3146], [191/840, 3151], [1/1680, 3212], [1259/5040, 3227], [1261/5040, 3229]}$ ${\tiny [1/5040, 3281], [1681/5040, 3316], [1679/5040, 3316], [1/2520, 3656], [2519/5040, 4141]}$

The final 5 are the bins adjacent to 0,1/2 and 1/3 (a bin for 1/2 would also be empty)