I don't think the Joe Smith deal would really impact how Garnett's tenure with the TWolves played out.
Let's say the Wolves never got caught, so they sign Smith to the promised $86m deal in the 2001 off-season when 7 year deals were still the norm. That puts him about ~$12m year, putting him around the top 15 of all NBA players at the start of the deal (though probably a little lower when annual raises are factored in). League wide luxury tax also goes into effect the next year. That small market ownership isn't paying luxury tax for that team. Wolves are probably worse off now. They probably don't trade for Sprewell because that would be taking on more money, they probably don't trade for Cassell (they actually traded a re-signed Joe Smith at ~$5m per year as part of the deal to get Cassell). Nobody in the league would trade for Smith's contract until it was an expiring. I just think that team would still be mediocre at best. If anything it affects the minor pieces (like C's acquiring Szczerbiak and using that salary for Ray Allen), but not KG wanting out around the same time he did.
Let's say the Wolves keep their draft picks. In any scenario (signing Smith or not) they're still a playoff team with a healthy Garnett early on, but just using their actual finishes, they would have had the following picks:
2001: 17 or 18 (depends on tie breaker), the 17-22 picks are Michael Bradley, Jason Collins, Zach Randolph (who they probably wouldn't have drafted because they had Garnett), Brendan Haywood, Joe Forte, Jeryl Sasser. Possible later gems include Gerald Wallace (25), Tony Parker (28), Gilbert Arenas (31), Okur (38).
2002: 23 or 24 (depends on tie breaker), the 23-29 picks are Tayshaun Prince, Nenad Krstic, Frank Williams, John Salmons, Chris Jeffries, Dan Dickau. If they won the tie breaker and got Prince, that would actually be a homerun for them. They have to win that tie breaker and make the right pick though. More interestingly would be how that affects the Pistons? A possible later gem is Boozer (36).
2004: 29. Picks 29 and following were David Harrison, Anderson Varejoe, Jackson Vroman, Peter John Ramos, Lionel Chalmers.
This is the team that drafted Ndudi Ebi (26) in 2003 while the next 3 guys drafted (Perkins, Barbosa, Josh Howard) all had 10+ year careers, and Rashad McCants (14) in 2005 ahead of Danny Granger (17). And in 2006 drafted Brandon Roy but traded him for Randy Foye.
This is not a team I'd bet on making the right pick, or being able to flip the pick for something useful to help Garnett (to be fair, it's tough even for a great GM to get value from a pick in the back half of the first round). In an alternate universe they have to not sign Joe Smith to the big deal and draft Prince and Roy, and flip Cassell/Sprewell for something of value to keep Garnett in Minnesota. I can't picture it.