I think it should. Some story things that should be done is fix some incosistencies that clash with the stuff in canon due to the later games in the series. Like Dante saying Mundus is the one who took the life of his mother and brother. When Kamiya made DMC1, Vergil was apparently kidnapped as a child. In DMC3, well, you get the idea.
Revoice it with the new actors- I want a serious Dante performance from Reuben Langdon near the end. I even want to hear that LIIIIIIIIIIIIIIIIGHT speech from him just because.
Add stuff from later games- For example, putting the style moves in for the 4 basic styles would be wonderful. Don't take any of the moves from the first game out, just add stuff from 3 and 4. Change the style system perhaps? And Dear God add a Bloody Palace(not time limit) and mission select screen!!!
Make the game even harder- Harder than even DMC3.
More cutscene that show off how over the top Dante can be?
Playable characters- I'd like to play as Nelo Angelo and Trish in Dante's levels. Maybe find a way to compromise the battle against Mundus since you have that ariel battle and you need Sparda DT's huge sword length to hit Mundus in the lava.
Keep the atmosphere the same- As you may know, DMC was suppose to be RE4. I want it to still feel like that but with all that other stuff I said earlier in it.
Bonus: Boss Rush. Hell, add some later game enemies like how you can fight Doku in NG3 RE.
I don't think Capcom would be really good at remaking it. They're struggling making new games as is, and they'll probably make it so "westernized" to attract the COD community that it will lose a lot of the charm and flair of the game. Either Hideki Kamiya remakes it, or no.