Logistically one difficult thing about a hypothetical offer sheet is that the original team gets a week to match. In most cases, the team would wait the entire week before officially notifying the NHL if only to keep the other team's cap space hostage.
The UFA goalie market tends to get settled quickly as it's a game of musical chairs and you don't want to be the one left standing. The past few opening days of free agency:
2022: Kuemper/Lindgren (WSH), Samsonov (TOR), Campbell (EDM), Halak (NYR), Comrie (BUF), Greiss (STL), Rittich (WPG), Copley (LAK), Jones/Hellberg (SEA), Add in Nashville signing Kevin Lankinen on the second day of free agency. Colorado/New Jersey added Georgiev/Vanecek at the draft.
2021: Ullmark (BOS), Grubauer (SEA), Andersen/Raanta (CAR), Mrazek (TOR), Reimer (SJS), Jones (PHI), Holtby (DAL), Elliott (TBL), Halak (BOS), Anderson/Dell (BUF), Broissoit (VGK), Bernier (NJD), Hutton (ARZ), Rittich (NSH), Pickard (DET). Colorado reacted quickly after losing Grubauer and traded for Kuemper.
So if you were to offer sheet Swayman on July 1, the rest of the goalie market could get settled while you're waiting on Boston to decide to match. If Boston matched, then you might have limited free agent or trade options by July 8.