I'd like to point out some corrections first
As far as I know, Bitcoin miners use a complex algorithm to verify each Bitcoin transaction and store it in the blockchain in the form of a block,
Miners dont directly verify transactions they include transactions in blocks and compete to solve a cryptographic puzzle which we call POW
Transaction verification is done by nodes in the Bitcoin network.
and since these are connected to many blocks, no illegal transaction can ever be validated
Bitcoin doesn't do any legality of illegality filter
As long as it obeys the consensus rule it would be valid
Except what you are talking about is transactions that doesn't meet the consensus rules.
And does Bitcoin depend entirely on miners?
No Bitcoin doesn't depend entirely on miners, but they are needrd for adding new blocks and security of the blockchain
Other vital parts are the Full nodes, users and developers.
And since Bitcoin mining prevents illegal Bitcoin transactions from being validated through Proof of Work, do you think there would be a chance for illegal transactions to occur if mining were to stop?
I will assume you mean invalid transactions and not illegal.
My response is No. Like you said without miners no new block would be created
and the absence of mining eliminates the possibility of any transactions being confirmed.
Invalid transactions still broadcasted would be rejected by full nodes regardless of mining activity.