Use scouts
Digging and building mineshafts is very easy to detect for even a casual observer. The BBEG can wait until the druid has wasted a bunch of his spells, then show up and bury the intruders in the grave they just dug. If it's an ambush, the PCs won't be able to teleport away without taking a few hits, and if they have a reputation for being flighty, the BBEG is likely to use poisons, diseases, curses, and other ongoing problems that won't be solved by a quick rest.
Use consequences for a slow approach
When the BBEG sees the PCs digging, he can predict where they'll end up and trap/reinforce that area of the lair. The PCs break into the lair, and it's flooded with neurotoxin or acid. This goes double if the PCs run away and then try to come back - the area will be swarming with minions, there will be defenses erected, and so forth. The BBEG may send his own spies out to find the PCs' base (or things they care about) and issue a retributive strike.
Use their reputation against them
Digging big ol' holes in the ground is far from business as usual in the realm of fantasy warfare. Stories of their weird tunneler types will spread, and evil overlords everywhere will defend their bases against this approach. The simplest solution is to bury explosive devices in multiple layers, so that the digging animals are killed and the druid needs to use another spell.
Use special terrain
Digging in dirt is easy. Digging through mud is easy, but maintaining the shaft afterwards is hard. Digging through granite is all but impossible. Digging through lava gets you dead. Digging into another plane or into a sky fortress makes no sense. If there's an underground river, digging underneath it becomes a challenge - you need to keep all that water out.
Use subterranean monsters and lairs
All those badgers are coming from somewhere. Some of them may have already done some digging - and instead of landing into the BBEG's lair, the party lands into a giant badger warren, or a fomorian hive, or the Underdark.
Use skill checks
Mine-shaft structuring is a complicated business; in real life, it's a job people go to college for. Does your druid have structural engineering knowledge? No? Then he put the support in the wrong place and the mine shaft collapsed.
Use third parties
There are more people in the world than the BBEG and the PCs. What if the PCs are digging in the middle of a sacred elven grove, and the elves come and ask them to stop? What if all their digging attracted another adventuring party, who decided to storm the citadel the old fashioned way and has already cleaned up by the time the PCs break in? What if the PCs are digging in crown lands, and a magistrate comes and demands to see their zoning permit?
Based off of DDAL1 modules...
The Table:
\begin{array}{l c l}
\text{Total GP Expended} & \text{Level} & \text{(Expenditure for next level)} \\ \hline
0 & 1 & (700) \\
700 & 2 & (1,000) \\
1,700 & 3 & (1,350) \\
3,050 & 4 & (1,850) \\
4,900 & 5 & (2,500) \\
7,400 & 6 & (3,500) \\
10,900 & 7 & (4,750) \\
15,650 & 8 & (6,500) \\
22,150 & 9 & (8,750) \\
30,900 & 10 & (12,000) \\
42,900 & 11 & (16,500) \\
59,400 & 12 & (22,500) \\
81,900 & 13 & (31,000) \\
112,900 & 14 & (42,500) \\
155,400 & 15 & (58,000) \\
213,400 & 16 & \\
\end{array}
The Method:
I went ahead and tabulated the possible XP to earn and possible treasure haul2 for each of the two-dozen or so DDAL modules I've got on hand.
Each module is designed for one of the following level spans3: 1-2, 1-4, 5-10, or 11-16.
From the XP earned in each I calculated a fraction of the indicated level span that would be "traversed" by completing the module, and used that to extrapolate how much gold would accrue to one gaining a level in that span.
I then weighted each by the recommended hours of play4 and ran a power regression.5.
Finally there's just a bit of rounding to make the numbers... round.
The Application:
Full confession: I don't think this ^^ is the best way to come up with these numbers.6 But I think it's a way and figured it's worth letting voters see so that wiser heads than mine can decide.
Those numbers are just... insane. Dropping that much cash--per adventurer!--onto the population of Barovia is just ridiculous. I know that D&D doesn't try to model any sort of functioning economy, but this is a bridge too far for my credulity. So you've got to find ways to ameliorate it. Perhaps use this table as a party table, so that once PCs have dropped 700gp everyone bumps to level 2? Perhaps adapt Delta's advice and switch to a silver standard (for XP) so that this all drops by an order of magnitude? Or perhaps...
magical items should be allowed as part of this scheme. Buying everyone a round and carousing for a night doesn't feel (to me) very different from rescuing a villager from weres and tossing a potion of healing their way. But this could also be a place where you can exert a GM's thumb on the scale, since the "values" of any items that the players disposed of in a way that buffed their renown aren't terribly standard. (And to that end, I'd recommend the Sane Magic Item Prices index; it may not be perfect, but it's waaay better than following the DMG's loose guidelines.) And now you've given players the interesting choice between hanging onto their widget of frobbing or "cashing it in" to level up. (Or level everyone up, per point 2!)
1 - Dungeons & Dragons Adventurers League, WotC's organized play program that was (and sort-of is?) active during 5e. The modules published through this program followed a rough set of guidelines for XP and treasure given out, and I'm using this as an insight into what WotC employees (Chris Tulach, specifically, perhaps with input from other designers?) thought was a good pacing of treasure accumulation by level.
2 - only coinage; since you indicate your players won't really be able to "cash in" any magical items, we're going strictly off of currency here. But more on this later.
3 - I'm using the term "span" rather than "tier" as "tier" is a defined term in 5e (PHB p.15) and these modules don't all correspond to tiers. It feels clunky, but I didn't want to conflate the ideas.
4 - Some modules--only in the lowest two spans--are recommended for two hours, most are recommended for four. This then has the effect of dialing back the impact of these smaller modules. However, there are many more of them (fifteen in 1-2 and 1-4) than in the higher levels (six in 5-10, one in 11-16) so the model's still drawing more of its info from those levels.
This weighting toward low-level information strikes me as fine in one way, as that's where the majority of play tends to happen. On the other hand, this means that any "errors" in the model are likely to appear at the high end, where it's going to take a while for you to notice and be really hard for you to fix: "oh, dear, we've been on level 10 for five sessions now, and are only halfway through. And I want them at 12 to face Strahd et al. for the last time and there's almost nothing left to explore!"
5 - power regression based on eyeballing this curve.
6 - I can think of one better way, but it's going to have to wait until I get my copy of CoS back from a buddy.
Best Answer
Let me summarize: You have a group that prefers Milestone XP, but isn't interested in creating any new stories. And you don't seem interested in making a story of your own unrelated to the characters.
The out-of-game options
Your players appear to be afraid that they will not be strong enough for the challenge you have prepared.
There are a few reason why they may think this way:
Either way, I suggest you warn them, out of game, that the game is not a videogame and that taking time to get stronger will have consequence on the world around. This will prevent resentment from growing if they don't expect the world to work this way.
The Session 0 option
If you think there is a major disconnect between the way you envision the game working and how you player do, either about the mechanical or the game difficulty or the fiction of adventurers of their power level taking on the challenges you have prepared.
The best option might be to go back to a session 0 and tell them "Why do you want to play? I don't want to make up a story unrelated to your characters, I'd rather it come from one of you. You can roll new characters if the current ones don't inspire you".
Maybe they will tell you they want to start fresh with a new kind of story, maybe they'll say they want to go along with what you have to tell. Either way, it's your turn to see if you want to GM for that game. Either way, letting someone else GM for a while is an option. So is just dropping the group (But it usually is the least happy one).
Or, why not roll with their idea?
They said their goal is to get stronger. Make them work for it, literally. Since you run Milestone, they litterally can't argue that they want to farm easy fights for experience. So they have no choice but to chase bigger and bigger beast.
Let them be monster slayers, wandering the land in search of bigger and bigger game. Solving problems because they happen to want to kill the source of the problem. You seem to have the perfect group for it. If you want to tell a more classic story as the GM, let the environment and the NPC's action tell it. Maybe one of your player will catch on to it and change the story again.
If you want to push the strategic game further. Or if you want to push the storytelling aspect of fighting epic monsters. Or if you were looking for a way to use the bigger monsters in the book: the Elder Red Dragons, the ArchDemilich or the godlike monsters at the end of the Monster Manual.
You have the perfect opportunity.