mainstream, equity-focused investors don't want to touch BTC or derivatives. they like to hold equities. there are very few affordable or liquid options to do that with that provide any sort of exposure to the crypto market.
i dunno about "kickstarting" a bitcoin bull run, but i could see a coinbase public listing drawing in attention/interest to the crypto space from traditional investors. it'll allow them to dip their toes into the market, get them comfortable with the idea of cryptocurrency as digital assets, and get them ripe for bigger listings like an eventual bitcoin ETF.
i think coinbase shares would be indirectly tied to bitcoin/crypto prices. the market would price shares based on earnings + speculation on future earnings. higher prices encourage adoption and more adoption encourages higher prices in a feedback cycle, all of which means higher earnings for coinbase. coinbase has huge market share, so what's good for crypto is good for them, and that's how the stock market will treat it.