How to sow wildflower seed?

A few points to consider is that the seeds need to be in contact with bare earth and that they need to be sown fairly thinly (which you did say)  Treading the area after sowing helps increase contact

The old fashioned hand operated spreader, called a fiddle, with the turning handle is excellent if you walk at an even pace up and down the area. Putting up marking posts at each end of each "walk" help you stay in a straight line and not overlap too much, once you know how wide the "fling" arc is. Sow once lengthways and then again crossways,: helpful to divide up the seed before you start, to ake sure you use the same amount for each section. This is how we sowed our daughter's neighbour's meadow which is 15m x 30m

One mistake we made was to not take out more grass than we did, by heavy raking after scarifying, so she had to thin it down a lot after the first season. Also grew some things in trays for her (cowslips and ragged robin I remember) which were planted out as smallish plants in one particular area rather than dotted everywhere

I think sand is too fine for those tbh, but it does help with actual hand sow as you can see where you have covered as you go.
The push along ones I have no experience of, sorry. I am sure someone will though

That mix sounds interesting... should look good. You will need to remove the mowings though as it will enrich the ground too much otherwise and the grasses will take over

My daughter' neighbour has two mown paths through her meadow so she can get to her pond, which is a sort of seep pond from the pasture above, and to the chicken run