I think there's a misunderstanding. Getting a seat on the board doesn't require "lots of shares" [in absolute numbers]. It requires "a large(ish) percentage of the voting shares".
That is, board members are selected by vote. If you have enough votes, you can vote yourself a seat on the board (or vote other board members off). Usually, no one has enough to do this unilaterally (more than 50% of the voting shares). Instead, investors with large percentages (5% or 10%, frequently) are selected because they have a large vested interest in the company and want to ensure that it is run correctly and in the direction that they prefer. At some interval, usually annually, all voting shareholders can participate in an election to choose the board (and do other things, sometimes). Since most investors will not attend, they assign their votes by "proxy" and someone else casts that vote in their place - which will almost always be for someone pre-selected by those with (relatively) large fractions of the ownership. In hotly-contested elections, getting enough of those non-attending shareholders can be a big deal, and this gives rise to the term "proxy fight", where competing factions are vying to be assigned proxy for enough of the minor shareholders.
Sometimes non-voting shares will also be granted board seats, but this is typically outside the regular channels.
