Can you run hot water... ?
You can hasten warm up with some well timed flushing, but a timer is a better answer. Since a really good electronic timer with lots of programmable turn on/turn off options is less than $20, why not?
An E-61 is a very good group head, but it's not the only good one. There are many fine machines which use other groups.
Let's start with your price range and whether you think you'd rather use a heat exchanger (HX) or electronically controlled double boiler (DBPID). There are pluses and minuses to each system.
In use, DBs are more "set it and forget it," while HXs require more focus, but do on the fly temperature adjustments. Because of the inherent complications of a DBPID, HXs tend to give you more construction quality for the dollar. By way of example, what costs around $1800 in an HX typically runs around $2300 in a DBPID.
Start by checking the purchasing FAQs here at HB, then run down the stickies and threads in the Buying Advice forum (also here). You'll find a lot. You'll also want to take a look at the Espresso Machine forum as there's a lot of experience and advice concerning bringing used machines up to snuff.
Good luck,
BDL