So despite it being pretty much the industry standard on properties with multiple interested parties you will willingly ignore advice from the expert you've hired and likely turn down tens of thousands of pounds of extra money? The method you've described won't drum up the same value of offers that best and final will and it's the reason agents use it - they work for you and ultimately want to get you the highest sale price.
It's hell for the buyer that's why.
Let's say I set my asking price at 250k. And I define two weekends in a row for viewing slots, say 6 per day so that's 24 slots give or take.
If there's loads of interest and I get say 5-10 offers (obviously assuming a buoyant market here) then I might consider I can get more money. And that's fine, and fair and I'm perfectly entitled to do that.
I wouldn't do best and final. I would approach each buyer and explain this:
"Thankyou for your offer on my property. There has been lots of interest and multiple offers around or above the asking price. I would like to invite you to consider increasing your offer and we will reconsider all offers when this is complete. Thank you."
Then when the next round of offers comes in, let's say they are all 10k ish higher.
By this time I will have got a feel for which buyers I prefer. For example one offer might be from a landlord who wants to rent the place out, and one from a family who wants to live there. I would approach the family and politely explain what's happened, show them the offers I have received and see if we can come to a deal everyone is happy with. I would be honest and transparent with them about what I'm looking for given the level of interest and offers on the table and give them the opportunity to properly think about it and respond, not be bullied into putting in a pie in the sky offer with no knowledge of other offers. They may not even have been the highest offer, but I would invite them to match the highest offer in that case.
Yes it might cost me a few k, but if the asking price was in the right ballpark in the first place then we won't end up far off the right value. And as long as I'm happy with the price and I can afford onward purchase then all is good.
If more people behaved like this we'd have a much less stressful buying experience. I have been on the opposite end of this as I said and won't do it to others, even if it does end up costing me a couple k.
Ps if the difference is tens of thousands, then the original asking price was set too low.