Why can't certain nucleoside analogs be used to label bacterial RNA?
Meng et al. actually give their explanation in the first paragraph of their results section. The main reasons for their claim are that they did not detect any labeling with 4SU or EU and that both analogs suppress bacterial growth. Also they raise the issue with the naturally occurring 4SU in tRNAs.

Aiming to develop nucleoside analogs compatible with bacterial RNA labeling, we first set out to evaluate the two nucleoside analogs widely used in eukaryotic cells. E. coli cells were incubated with 4SU or EU at varied concentrations for 2 h, which resulted in no detectable labeling even at high concentrations (Supplementary Figure S1A, B). Moreover, severe suppression of bacterial growth was observed with both 4SU and EU (Supplementary Figure S1C, D), which was in agreement with previous observations that 4SU impaired cellular metabolism in E. coli (31). Cell viability was significantly impaired with 4SU at concentrations higher than 100 μM, while EU did not induce significant decrease in cell viability (Supplementary Figure S1E, F). In addition, tRNAs of bacteria contain endogenous 4SU generated via thiolation of uracil (32), which may interfere with metabolic labeling if exogenous 4SU would be used. Of note, although EU does not incorporate into DNA in mammalian cells (10), significant labeling of DNA was observed in EU-treated E. coli, presumably due to metabolic conversion of EU to EdUTP (Supplementary Figure S1G). Together, these results demonstrate that 4SU and EU are not applicable for bacteria.

This explanation is not necessarily incompatible with the results in the other references mostly because the strains and the test approaches are different.
In Favre et al. they use a different strain of E. coli,&quot;an Escherichia coli
pyrD strain&quot; that is optimized for 4SU incorporation :
In Material and Methods :

pyrD used to control the level of 4-thiouridine incorporation

and in Results :

a) growth of E. coli AB1157 sfiA pyrD in the presence of 4-thiouridine
Preliminary studies showed that the level of s4U incorporation into the rRNA of strains MRE 600 or K12 AB1157 grown in a standard glucose medium in the presence of 4-thiouridine remained at the detection limit. We therefore use AB1157 pyrD mutant deficient in the pyrimidine biosynthesis pathway. To our growth medium we systematically add 40 μg/ml uridine because it is  the minimal concentration required to supplement growth fully. The growth of the pyrD mutant was first examined at different temperatures and different exogeneous s4U concentrations from 40 μg/ml upwards in the conditions defined in Material and Methods, In Fig. 1 we report  the growth curves obtained for two s4U/U ratios,1 and 2.5, and four temperatures: 25°C, 28°C, 30°C, 35°C. When the ratio is 1 there is practically no growth inhibition. With  a ratio of 2.5 the  growth  inhibition becomes strongly dependent on temperature. The culture does not grow at all at 20°C, the  growth is linear between 25°C and 30°C and for temperatures above 30°C it escapes inhibition, becoming exponential. As will be described below significant s4U incorporation occurs in  conditions where growth is linear.

In Favre et al. as seen at the end of the paragraph above as well as in another publication (Bezerra and Favre, 1990) from the same lab shows that incorporation of 4SU depends on several factors.
Concerning the work of Blenkiron et al., it is harder to explain but they also use a different strain so that may impact the incorporation and they do not use the same approaches compared to Meng et al. to detect it. Confocal microscopy for Blenkiron et al. vs Dot blot and In-gel fluorescence for Meng et al..
Finally as far as the grant abstract is concerned, it's hard to conclude because the information is scarce but as Meng et al. they also observed an effect (although milder) of 5EU on bacterial growth and again, they used a different approach to observe the labeling based on fluorescence microscopy as Blenkiron et al..

References :

Meng L, Guo Y, Tang Q, Huang R, Xie Y, Chen X. Metabolic RNA labeling for probing RNA dynamics in bacteria. Nucleic Acids Res. 2020 Dec 16;48(22):12566-12576.
Favre A, Bezerra R, Hajnsdorf E, Lemaigre Dubreuil Y, Expert-Bezançon A. Substitution of uridine in vivo by the intrinsic photoactivable probe 4-thiouridine in Escherichia coli RNA. Its use for E. coli ribosome structural analysis. Eur J Biochem. 1986 Nov 3;160(3):441-9.
Blenkiron C, Simonov D, Muthukaruppan A, Tsai P, Dauros P, Green S, Hong J, Print CG, Swift S, Phillips AR. Uropathogenic Escherichia coli Releases Extracellular Vesicles That Are Associated with RNA. PLoS One. 2016 Aug 8;11(8):e0160440.
Bezerra R, Favre A. In vivo incorporation of the intrinsic photolabel 4-thiouridine into Escherichia coli RNAs. Biochemical and biophysical research communications. 1990 Jan 15;166(1):29-37.