@awnlee jawkingAssuming your pieces are single sided, print the same design on both sides (so they can be either way up)
Assuming not all squares are to be covered by pieces, include blank pieces so all squares will be covered
Have the board as a tray with sides nearly the same height as pieces are thick
Shake all pieces together in a bag/box/tin, then pour then into the tray
Gently shake the tray, including raising one corner so all pieces register to the opposite corner or slide off the tray if they were stacked on others
Collect the pieces that fall off, and shake these again, then pour into the empty area
Repeat the last two actions until there is only one piece not placed
Put last piece in remaining hole.
The last few rounds of this could be sped up by arranging the tilt to be uneven in the two directions so there is preferentially just a single row vacant in the latter stages, then pouring the remaining pieces out in a line adjacent and transferring them across
The tray could be made double sized (10*20) with a transparent cover over one half (with enough height/gap for just one piece to slide in) and higher walls on the other half, then pour once and shake until all pieces have migrated under the cover
The oversize tray extension could be made into the tin/box by extending the side height and adding a cover; tip the board one way to slide the pieces into the box, shake, tip the other way and shake until all pieces have fallen into the board (which is still the 1-high, covered tray)
If the tray extension is the same size as the board, and somewhat higher than a piece on edge (to enhance the randomisation shaking), it could be hinged at the junction to allow it to be folded under as a kind of plinth during play leaving a single board sized, unobstructed playing surface. The problem of sealing the transfer slot whilst not transferring pieces to the board is left as an exercise for the astute reader (I've thought of a few)
The hinge area could be covered with a curved loading ramp to guide pieces into the board with the board and plinth at an angle (perpendicular?). This curved feature could be reproduced on the other edges of the board to provide a consistent look. The plinth/box would probably need a tapered funnel structure to feed pieces into the curved guide.
If the plinth height was increased, it could probably incorporate a full width taper with a nearly 180 degree loading ramp so the hinge could be eliminated, but this would probably look unbalanced and unwieldy.
Alternatively, have a single sized tray with a lid deep enough to form a box, shake vigorously to mix then gently to spread out, continuing the last until a single layer is formed.