First, yes, typo, Chase points can be transferred to United.

Second, we're talking about "saver" awards. Again, the consensus on FT would be that full-price awards are not a very good use of points.

On June 2, there is a Lufthansa flight (bookable with United miles) leaving DTW at 7:10PM and arriving FRA at 9:20AM, then a Lufthansa flight leaving FRA at 1:50 and arriving ATH at 5:35, for 50K miles and $36.10 in fees. I didn't look at return trips, but there are 50K tickets available also on June 3, 4, and 5.

Also, I'm not sure of all the rules, but often the cash-and-miles upgrades apply only on the operating carrier. So if you buy a ticket on a Lufthansa flight that is codeshared with United, there's a very good chance you cannot upgrade.

Also you miscalculated I think, you noted 20K points + $500 each way, that's a total of 80K points you will need.

As for the United bonus, there are different bonuses depending on how you do it. The general signup bonus is (I think) 25K, plus 5K more for adding an authorized user, plus 10K for doing something else that is not easy like making a big balance transfer or spending $25K in the first year. So it's really more like 30K. However, there are links available for 50K+5K. To get this, you generally have to (a) have at least one mile in your United account (so open a United account and transfer some Chase points into it) and (b) be logged into your United account. Read the detailed thread on this card for more information.

I understand that you have to work with what you have. But if you value Chase points at 1.5 cents each (that's in the range) then a paid ticket plus upgrade would be $1500 for the ticket, $600 worth of points, $1000 in cash, or $3100. A points-only ticket would be $1500 in points plus about $150 in fees, or half the price. It's really worth it to jump through some hoops to scrape together the extra points or miles.