There are multiple models for price impact and the one you have listed here is not the latest. You can see a writeup of a few of the most popular and recent models in this answer.
We can think of a few reasons why price impact is considered linear in the bid-ask spread.
First, you want the trade to be completed (implied by most of these models). You may be unable to wait for your price to be taken; you will need to switch from a price maker to a price taker, cross the spread, and trade at the far side fo the market. In that case, you pay the bid-ask spread to guarantee execution. However, that is not certain to occur, so $\alpha$ accounts for that probability.
Second, suppose you trade in dark pools (or other ATSs using occasional auctions/matching at the midpoint). The imbalance of orders in a matching auction will bleed back out to the market and so instead of trading at mid-market when you entered your order, the price will shift a little. In that case, you again end up paying a price that is half the spread plus or minus some shift in that midpoint.
Third, market makers who take on a position when you trade with them eventually need to hedge. Doing that may require them to immediately get out of a position, so they pay the bid-ask spread to trade immediately. Their quotes will therefore reflect the spread to pass that cost on to you.
If you have not traded much, you might want to try doing so in a good simulator or (far better) with some real money. You will quickly see how often you need to consider crossing the spread and why that affects the price you pay.
You should also probably read up on permanent versus temporary (and decaying) price impact at the link above. That may help clarify the thinking of how the bid-ask spread affects prices.