We're still in a bit of a 'just coming out of lockdown' phase which makes things a bit awkward and yes, you will have to book even for drinks right now. Curfew is 10.30 pm too! Hopefully this all changes by July but cases are really high in Edinburgh right now...
Anyway. Drinks - Bramble, Panda & Co, Hoot the Redeemer.
Food and drink - Lady Libertine, Bon Vivant
Casual - The West Room (10 mins from your hotel)
Breakfast - Dishoom, always. Or walk down to Stockbridge from your hotel (about 20 minutes down the Water of Leith) for good cafes
Other food - BABA is great and close to you, also Noto. Aizle also great but very full-on tasting menu type thing. The Little Chartroom if you can get in.
If you go east to Broughton Street you can have a merry wee time of it - margaritas in The Basement (bit of an Edinburgh institution), food at Taistel or the Ox and there's a new bar called Broughton or The Broughton (i can't remember because I haven't been out in forever!)
If it's sunny... there's no point in giving you any beer gardens because they're all booked out till Christmas but try the Holyrood Distillery. The places along George Street aren't my favourite (apart from Contini where you'll also get a great brunch) but they have outdoor seating and most of them keep spaces for walk-u ps.
Actually though your concierge may also be a good shout at the Waldorf - the fun is discovering new places yourself normally but those spontaneous days aren't quite back with us yet.