To a first order approximation, I’d say bidding wars don’t change the block inclusion fee to any significant degree. Whether a bot bids 20,000 Gwei instead of 100 Gwei, it still consumes the same amount of block space.
It’d be pretty rare for more than two bidding wars in a single block. (The major exception is when the price of ETH/USD swings exposing many separate Defi arbitrages at once.) A Uniswap trade with a gas refund uses about 60,000 gas out of the 12.5 million block limit. So “ordinary” transactions almost never have to worry about competing with the gas bid at the top 1% of the block.
The block inclusion, or even median fee is not meaningfully affected. In the same way that billionaires competing to buy beachfront property in Oahu doesn’t meaningfully increase the price of housing in Peoria.
Bidding wars may potentially decrease the gas impact of front running. If auctions never get too expensive than many marginal front runners would be tempted to make an attempt. The lower the gas fee the more a free option it is. (Even if you lose the bidding war a failed Uniswap trade costs about 26,000 gas.) Fierce bidding wars probably drive marginal players out, which frees up block space, lowering prices at the bottom of the block, despite increasing prices at the very top.
That being said, bidding wars may have an indirect impact through gas oracles. Hopefully gas oracles use non-parametric statistics, like median. But imagine a gas oracle that uses arithmetic mean. It sees 100 transactions at 100 Gwei and one at 20,000 Gwei. Then it tells its users to bid 300 Gwei to get mined. If enough ordinary users were using that gas oracle, then even a small right tail of super-high gas trades could indirectly raise everyone’s costs.