Well, you have to differentiate between Vanguard as a broker and Vanguard funds.
As far as funds go, Vanguard (Jack Bogle) created the first retail index fund and index funds have always been a big part of the business. These have been conventional mutual funds. ETFs came later, I believe after Bogle retired as he didn't like them. Vanguard has historically been the low price leader, primarily because it is a mutual company/owned by the fund shareholders, so there is no profit taken by a management company. In the past couple of years the brokerage houses have been slitting each others' throats on fees to the point that many total market funds have negligible or no fees. IMO VG has lost most of their cost advantage. But I don't think we've seen the end of the fee battle; these companies have to make money somewhere. It's IMO a watch and wait situation. We are heavily into VG funds.
As a broker, Vanguard gets mentioned together here with Schwab and Fido (Fidelity). All seem to be viewed as white hats with only slight differences. Vanguard has only two or three offices around the country where the other two have many. Face-to-face is nice sometimes though I don't see my Schwab guy more than once in a couple of years. AFIK, Vanguard's telephone relationships tend to be more impersonal. They do offer some cheap management options though. They responded to a small nonprofit's RFQ that I wrote and offered to run $4m for 40bps but with a very impersonal interface. They were the low bidder but we passed because we wanted a little more personalized service. As it turned out that was the right decision. IIRC we are paying 59bps now.
You can buy Vanguard funds at Schwab and Fido, sometimes with a small transaction fee. If you are buy and hold as we are it is completely negligible. So wanting VG funds doesn't need to sway your choice of a broker. VG may not have as broad a fund offering as Schwab and Fido, but you should check as I am guessing.
I'm surprised you have a two week decision deadline. Normally an account can be transferred between brokers or custodians at any time by simply filling out a form at the receiving company and transferring the assets in kind. Very low friction and usually fairly fast.
Good luck.