Seller wants to stay on after exchanging contracts

[asks62]

A bit of a strange one I must say in our journey of finding our first home. We've had a few difficult scenarios thrown at us in the last few months but this one, I must admit is a bit too crazy to digest. Looking for any advice you can offer.

Offer accepted. We're in a rush obviously to beat the Jun 30 stamp duty deadline. (It's a big sum for us if we were to delay).
Few days later the estate agent says the sellers are doing up their new property and it'll take a few more months than planned earlier. So they're still happy to close the transaction by end of June but they will stay over for 3 months beyond the contract completion date.
They're happy to pay rent for the 3 months that they stay at technically what would be our property if we go ahead.

Problems:

1. Solicitor and bank have both red signalled it saying you need to take possession on contract exchange day. So essentially I've to lie to them saying I will be moving in there as soon as the contracts are exchanged. I'm extremely uncomfortable with this.
2. Since we would be owners all liability on the house is ours. Building insurance etc could be invalid if something were to go wrong like a fire, burglary etc.
3. We love the property. It's a nice area but we're too inexperienced to visualise the challenges we might get into because this could be breach of contract with lender etc.

The sellers seem genuine but I know nothing about them.

There are not many properties on the market because everyone seems to be buying one at the moment.
My mind says we should just let it go.

Is this a common scenario? Do you think it's worth the pain?
Estate agent is trying to talk me into it and I'm confused.
Any advice is appreciated.

is it common ?
would those insurances be valid if needed, aren't they for owner occupiers.
so insurer could decline cover if something amiss happened as nether party were the occupiers of the insured property a the time of the incident.

Yes it is common to have a gap between exchange and completion and the new owner needs to have insurance to cover from exchange. Most insurance companies cover this period as a specific clause in the agreement, it isn't a separate type of policy. Some stipulate a maximum period for exchange to completion - usually 30days.

A hard no to staying after completion. Given they've been so cheeky I wouldn't even go for an extended time between exchange and completion because I couldn't trust them to stick to it.
When we sold, our buyers tried to get us to drop the price by 20k on exchange day. We said no and that we'd put it back on the market if they couldn't pay the agreed price. We did manage to sort something out in the end, but it still rankles.
Don't do it.