Index ETFs an Ex Divided Dates - Why do ETFs fall on Ex Div date?

Today this ETF went Ex -Div: HSBC FTSE 100 ETF HUKX.L

It dropped by the amount of the FTSE decline (tracking), plus it’s own ex-div amount.

So far, so good.

However – The instrument is designed to track the FTSE 100 – the underlying instrument (FTSE 100) is adjusted for dividends at other times, and loses value on some Thursdays when it’s it’s constituents go Ex div themselves.

So doesn’t the ETF suffer the effect of ex-div twice – Both when itself goes ex-div, and when the underlying index it’s tracking loses value due to ex-div activity?

How does the ETF value catch back up with the FTSE value to track it?

Is this because we trust the fund manager to reinvest the dividends they receive into the fund from the underlying investments?

Is the fund manager obligated to reinvest the dividends received immediately?

For timing purposes, shouldn’t there be some instances where the ETF value is greater than FTSE 100, because some dividends of underlying stocks have been collected and reinvested, but the ETF itself hasn’t disbursed them?