I'm not sure there's any way to convincingly argue for Budge, there's definitely a lot of gray area there. But here goes.
1941 is a strange case, and I'm definitely not an expert on it. But from a quick lookup here's what I'm going off of: Budge won his tour with Tilden convincingly, then only played in three tournaments and didn't win any. The guy had a streak of 10 Slam wins in 11 attempts from 1937-1942, losing only the 1941 match, I'm willing to give him pass, especially with his dominance in both 1940, 1942, and on the Tilden tour in 1941. Very, very possible I think that Budge was the best player in the world that year, and small sample sizes just occasionally produce strange results like that.
I think Budge has a decent case for world #1 from 1937-1942, with main accomplishments listed below. Again, not 100% on their accuracy, but it's what I was going off of in making the ranking.
1937: Budge wins both Amateur Slams he enters
1938: Budge wins the Grand Slam (meaning he's won 6 consecutive Amateur Slams). Turns pro.
1939: Budge def Vines 22-17
1939: Budge def Perry 28-8
1939: Budge def Vines 15-5 (in a separate tour)
1939: Budge wins Wembley Pro Slam RR (3-0 vs Nusslein, Tilden, Vines; dropped just one set to Nusslein)
1939: Budge wins French Pro (without dropping a set in any round; beat Vines in Final)
1940: Budge wins US Pro (drops just 1 set over four matches, to Perry in the Final)
1941: Budge def Tilden 46-7
1941: Budge loses first round at US Pro to John Faunce (straight set loss; no idea of the context)
1942: Budge easily wins round robin tour with 52-18 record over group of Riggs, Perry, Kovacs, Stoefen
1942: Budge wins US Pro (without dropping a set in any round; beat Riggs in the Final)
Ok just looked up Budge's loss to Faunce in 1941. It came the day after he got married
. LOL. Only Slam loss in a 6 year period comes the day after his wedding. (We need more digging to find out his blood alcohol content during that match.)
Anyway, it's pretty unheard of for an Amateur to turn pro then immediately knock off the world's best player (and a quality one in Vines) by a tour H2H of 37-22. And unless you're giving a massive amount of credit to three tournaments in 1941 over the hundreds of tour H2H matches he played, I think it's safe to say Budge was the likely clear #1 from at least 1939-1942, with a good argument in 1937 and 1938.
So that's pretty good, and unlike Borg who left on his own terms, or Kramer who had injury issues, Budge's dominance got interrupted by mass genocide. Again, willing to give him a little bit of a boost there, as he likely pads his numbers a bit (and doesn't hurt his shoulder) if it weren't for the, you know, mass genocide. Then he comes back, still top-5 through 1949, still making Pro Finals until 1953 (his ranking into the 1950s is murky, I agree).
Lots of subjectiveness in that analysis for sure. But compare that career to Borg's. Borg was dominant on two surfaces, but clearly had his rivals and never won the US Open. I'm not sure Budge had a rival during his prime. And while Borg's career as an elite player spanned 8 years, Budge's spanned at least 13 and possibly a bit more.