If you just keep putting off dealing with the snow, it will eventually just melt on its own
You have had good suggestions so far.
Dead centers are more precise than live centers, but a good live center will give the same results as a dead center unless you are holding under 5 tenths tolerance. You can get live centers this good, but they are not worth it IMO unless you are doing a lot of work like that.
I use a lube meant for dead centers, and it works really well. Just search for "center lube" from who ever you buy supplies from. It has more in common with hypoid gear oil than cutting oil - it needs to both bear pressure and sliding action.
Make sure you use a center drill, not a spotting drill. The drill should have a small diameter tip coming back to a 60* shoulder and the larger diameter. The drill you described sounds like a center drill. The little tip on the drill serves two purposes: it prevents the center from bottoming out on it's tip, and is a reservoir for oil. Just get a good dab of oil into the hole and put some on the center and it will be fine. If you will be using the dead center a lot, it is worth investing in quality center drills, and possibly a center hone. The better the finish on the center hole, the better it will center and the longer the center will last. I sometimes just put the corner of a fine slip stone or 320 paper into hole and polish it a bit before using the dead center (make sure you clean the hole out afterwards). Don't make the hole too perfect, you need some scratches for the oil to go in, you just want to polish off the chatter and burs.
The center should be just tight enough to support the work, but not so tight it is oil starved. The work should ride on a film of oil. With a little practice, you will get the feel for it. The ram will have sort of a sticky feel to it when you withdraw it as you over come the surface tension of the oil.
I doubt the problem you had is with the live center. Even the cheapest live center should not wobble. One of my centers is a medium duty live center from Grizzly, it works just fine and I use it often. More likely the work just did not go back into the lathe the same as it came out. This is common, and even more common with 3jaw chucks. Generally speaking, people do not remove work from a 3jaw chuck if they need to put it back exactly where it was. A really good chuck will get you really close (you may have paid less for the lathe than a good chuck), a cheap chuck will get you in the same country. If you need to remove the work and put it back in, there are a few options. Use a 4jaw and put it back reliably. Get a good "set true" chuck. Turn between centers (this is the most reliable way).
You can get it back in pretty well by putting marks on the chuck and work like you did. Add to that running the center up into the work before tightening the chuck and you will be even closer. You said you cut a taper on the work. Did you cut the taper by moving the tail stock off-center? I am not sure it would ever be possible to put the work back in a chuck on center if this is so. You should also use a live center when possible do it this way. If you do have to use a dead center, you will have to drip lube on it very frequently.
I was once a proponent of tightening all the keys on my 3jaw. The fine folks here set me right on this subject, you should use just the same key every time. 3" x 10" is a bit much for a 5" 3jaw and is pushing the limits of your lathe a tad. I would suggest using the 4jaw for work this massive in general. I just don't see being able to reliably drill a center hole with a piece of stock that large hanging out in the wind on a chuck that small. The 4jaw will grip much tighter and allow you to get it properly centered for drilling.
