I think it is fine to let him go now. The short-term gain is to at least give players who were thinking about leaving in January, some pause to re-consider. Players who might have felt stifled under Mourinho.
The caretaker manager can also try new ideas, which Mourinho seemed to be running out of.
And I think fans all want Woodward out as well... but not likely under the Glazers cos Woodward is a finance guy who has kept the $$ coming in and stock price high.
My personal choices, in order of preference:
Diego Simeone - has been managing Atletico Madrid since 2011 and actually broke RM/Barca's dominance to win La Liga and run both of them close on another ocassion (finishing 3rd despite amassing 88 points). He also won two Europa Leagues and got them to two CL finals. I especially like that he did so while losing star players like Falcao (who was still playing well at the time), Costa, Courtois, etc - and has shown longevity and loyalty. Currently 3rd (on goal diff only), 3 pts behind leaders Barca.
Mauricio Pochettino - has the experience in the premier league and managed to make Spurs a team capable of challenging for the title, despite having a scrooge of a boss like Daniel Levy. Has all the potential but just a small gripe that he hasn't actually won anything. His win ratio with Spurs is 56.4 - still below Mourinho's 58.3 with Utd.
Zinedine Zidane - great record he might have with 3 consecutive CL titles and a La Liga title, but he inherited a squad brimming with superstars and an in-form CR7. It is also a very short career and I am not entirely convinced he can sort things out at Utd. His abrupt quitting also counts against him, but one can hardly blame him with a boss like Florentino Perez who fires coaches even after they win La Liga. He might be the man-motivator that Utd exactly needs right now, but then he might not be and he has not shown he can build a squad from (almost) scratch.
Antonio Conte - won the league title and FA Cup in two seasons with Chelsea. Has premier league experience and proven he can win, especially after inheriting a squad from Mourinho...
Strict disciplinarian like Fergie was, and no lack of passion pitchside. Perhaps a good choice as caretaker manager.
Steven Gerrard - leading Glasgow Rangers to the title...... just kidding lah... wahahahaha....